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Basic CalculusQuarter 3 – Module 5:The Derivative of an Algebraic,Exponential, Logarithmic, andTrigonometric Function11Basic Calculus – Grade 11Quarter 3 – Module 5: The Derivative of an Algebraic, Exponential, Logarithmic, andTrigonometric FunctionFirst Edition, 2020Republic Act 8293, section 176 s...

11 Basic Calculus Quarter 3 – Module 5: The Derivative of an Algebraic, Exponential, Logarithmic, and Trigonometric Function

Basic Calculus – Grade 11 Quarter 3 – Module 5: The Derivative of an Algebraic, Exponential, Logarithmic, and Trigonometric Function First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education – Region XI Regional Director : Evelyn R. Fetalvero Assistant Regional Director: Maria Ines C. Asuncion Development Team of the Module Writers: Genelyn A. Barbasan Editors/Reviewers: Steve G. Zacal Illustrator: Layout Artist: Template Developer: Neil Edward D. Diaz Management Team: Reynaldo M. Guillena Jinky B. Firman Marilyn V. Deduyo Alma C. Cifra Aris B. Juanillo Antonio A. Apat Printed in the Philippines by ________________________ Department of Education – Region XI Office Address: Telefax:

DepED Davao City Division, E. Quirino Ave., Davao City, Davao del Sur, Philippines (082) 224-0100

11 Basic Calculus Quarter 3 – Module 5: The Derivative of an Algebraic, Exponential, Logarithmic, and Trigonometric Function

Introductory Message For the facilitator: As a facilitator, you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their learning at home. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module.

For the learner: As a learner, you must learn to become responsible for your learning. Take time to read, understand, and perform the different activities in the module. As you go through the different activities of this module be reminded of the following: 1. Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises. 2. Don’t forget to answer Let Us Try before moving on to the other activities. 3. Read the instructions carefully before doing each task. 4. Observe honesty and integrity in doing the tasks and checking your answers. 5. Finish the task at hand before proceeding to the next. 6. Return this module to your teacher/facilitator once you are done. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain a deep understanding of the relevant competencies. You can do it!

ii

Let Us Learn

This module was designed and written with you in mind. It is here to help you master the concept of the derivative of an algebraic, exponential, logarithmic, and trigonometric function. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. The module is composed of one lesson: ➢ Lesson 1 – The Derivative of an Algebraic, Exponential, Logarithmic, and trigonometric Function After going through this module, you are expected to: 1. determine the relationship between differentiability and continuity of a function (STEM_BC11D-IIIf-1); 2. apply the differentiation rules in computing the derivative of an algebraic, exponential, logarithmic, trigonometric functions and inverse trigonometric functions (STEM_BC11D-IIIf-3).

1

Let Us Try Vocabulary Check: Match the column B to its corresponding definition in column A. Column B

Column A

_______ 1. The use of a raised number to denote repeated multiplication of a base. _______ 2. A point at which a function is not defined. The graph has a break at this point.

a. b. c. d. e.

Discontinuity Limit Exponent Zero Differentiable

_______ 3. The derivative of a constant function. _______ 4. A value to which a sequence or function converges. _______ 5. A function that has a defined derivative (slope) at each point.

Let Us Study Differentiability and Continuity of a Function A function is continuous if we draw without ever lifting our pen from the paper. We can also say that the graph of a continuous function has no holes in it. Let us consider the three graphs below where all three functions have a limit at x = 2, and then work to make the idea of continuity more precise.

l

m

n

The figures show the functions l, m, and n has different behaviors at x = 2. 2

Note that l(2) is not defined, which leads to the resulting hole in the graph of l at x =2. It means that l is not continuous at x =2. For the function m, we observe that while lim 𝑚(𝑥 ) = −3, the value of m(2) = -2, and thus the limit 𝑥→2

does not equal the function value. We say that m is not continuous, even though the function is defined at x = 2. Lastly, the function n appears to be the most well-behaved among the three, where at x = 2 its limit and its function value agree. That is, lim 𝑛(𝑥 ) = −3. With no hole at x =-3, then it is 𝑥→2

continuous.

By the definition, a function is said to be continuous at a point x = c, if lim 𝑓(𝑥 ) exists, and lim 𝑓(𝑥 ) = 𝑓(𝑐). It implies that if the left-hand limit (LHL), 𝑥→𝑐

𝑥→𝑐

right-hand limit (RHL) and the value of the function at x = c exists and these parameters are equal to each other, then the function f is said to be continuous at x = c. A function is said to be differentiable at the point x = c if the derivative f’(x) exists at every point in its domain. 𝑓(x + h) − 𝑓(𝑥) ℎ→0 h

𝑓′(𝑥) = lim

Example 1. Relationship between Differentiability and Continuity of a Function Consider the function 3 differentiability at x = 4.

f(x)

=(4𝑥 − 3)1/4.

Solution

LHL

𝟑

lim− = (𝟒 (𝟒) − 𝟑)𝟏/𝟒 = 0

𝑥→3/4

RHL

and

Thus, the function is continuous 𝟑

at about the point x = 𝟒.

𝟑

lim+ = (𝟒 (𝟒) − 𝟑)𝟏/𝟒 = 0

𝑥→3/4

continuity

LHL = RHL = f(c) = 0.

lim = (𝟒𝒙 − 𝟑)𝟏/𝟒

𝑥→𝑐+

its

Explanation For checking the continuity, we need to check the left-hand and right-hand limits and the value of the function at a point x = c.

lim = (𝟒𝒙 − 𝟑)𝟏/𝟒

𝑥→𝑐−

Discuss

𝑓(x + h) − 𝑓(𝑥) ℎ→0 h

𝑓′(𝑥) = lim

Check the differentiability at the given point.

𝟑 𝟑 𝑓 ( 𝟒 + h) − 𝑓(𝟒) lim h ℎ→0

𝟑

Substitute x as 𝟒.

3

lim 𝑓 ([4 (𝟑𝟒) + h] − 3) ℎ 0

1/4

h

𝟑) − 𝟑)𝟏/𝟒 − (𝟒 (𝟒

(𝒉)𝟏/𝟒 − (𝟑 − 𝟑)𝟏/𝟒 lim ℎ→0 h

Using composite function, input the x to f(x) in the numerator. Subtract the numerator

𝟏 Simplify by subtracting exponents (𝒉)𝟒 − 0 of h. lim ℎ→0 h 1 Thus f is not differentiable at x = lim 𝟑 = ∞ 𝟑 ℎ→0 . 𝒉𝟒 𝟒 We see that even though the function is continuous but it is not differentiable.

Differentiability, continuity, and existence of a limit are also called the local properties of a function. This means that a function may be differentiable at one point, but fail to be differentiable at a different point; similarly, a function may be continuous or have a limit at one point, but not continuous or have a limit at another point. A function f is continuous at x = c whenever f(c) is defined, f has a limit as x →c, and the value of the limit and the value of the function agree. This guarantees that there is not a hole or jump in the graph of f at x = c. A function f is differentiable at x=c whenever f ‘(c) exists, which means that f has a tangent line at (c, f(c)) and thus f is locally linear at the value x =c. Informally, this means that the function looks like a line when viewed up close at (c, f(c)) and that there is not a corner point or cusp at (c, f(c)). The derivative of an algebraic, exponential, logarithmic, trigonometric functions and inverse trigonometric functions The Basic Rules The functions f(x) = c and g(x) = xn where n is a positive integer are the building blocks from which all polynomials and rational functions are constructed. To find the derivatives of polynomials and rational functions, we must know the formulas for differentiating the basic functions. Constant Rule Let c be a constant. If f(x) = c, then 𝑓’(𝑥) =0. Alternatively, we may express this rule as 𝑑

𝑑𝑥

(𝑐 ) = 0

or

4

𝑓′(𝑐 ) = 0

Example 2. Find the derivative of 𝑓(𝑥 ) = 5 Solution: 𝑓′(𝑥 ) = 0

Derivative of constant number is 0

Power Rule Let n be a positive integer. If 𝑓(𝑐 ) = 𝑥 𝑛 , then 𝑑 (𝑥𝑛 ) 𝑑𝑥

= 𝑛𝑥𝑛−1 or 𝑓′(𝑥) = 𝑛𝑥𝑛−1

Example 3. Find the derivative of 𝑓(𝑥 ) = 𝑥 2 Solution: 𝑓′(𝑥 2 ) = 2𝑥 2−1 𝑓 ′(𝑥

2)

The power of x becomes the coefficient of the term and subtract the exponent by 1, then simplify.

= 2𝑥

Constant Multiple Rule The derivative of a constant k multiplied by a function f is the same as the constant multiplied by the derivative.

that is,

𝑑 (𝑘𝑓(𝑥)) 𝑑𝑥

𝑑

= 𝒌 𝑑𝑥 (𝑓(𝑥))

ℎ′(𝑥) = 𝑘𝑓 ′ (𝑥)

Example 4. Find the derivative of 𝑓(𝑥 ) = 6𝑥 3 Solution: 𝑓 ′ (𝑥) = 3(6𝑥 )3−1 𝑓 ′ (𝑥 ) = 18𝑥 2

Multiply the constant coefficient to the exponent, subtract the exponent by 1. then simplify.

Let f(x) and g(x) be differentiable functions and k be a constant. Then it holds the following equations. Sum Rule The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.

that is,

𝑑 (𝑓(𝑥) + 𝑔(𝑥)) 𝑑𝑥

𝑑

𝑑

= 𝑑𝑥 (𝑓(𝑥)) + 𝑑𝑥 (𝑔(𝑥))

ℎ′(𝑥 ) = 𝑓′(𝑥 ) + 𝑔′(𝑥 )

5

Example 5. Find the derivative of 𝑓(𝑥 ) = 3𝑥 2 + 5 Solution: 𝑓 ′(𝑥) = 2(3𝑥 )2−1 + 0 𝑓 ′ (𝑥 ) = 6𝑥

Apply the constant multiple rule, add, then simplify

Difference Rule The derivative of the difference of a function f and a function g is the same as the difference of the derivative of f and the derivative of g.

that is,

𝑑 (𝑓(𝑥) − 𝑔(𝑥)) 𝑑𝑥

𝑑

= 𝑑𝑥 (𝑓(𝑥)) −

𝑑 (𝑔(𝑥)) 𝑑𝑥

ℎ′(𝑥) = 𝑓 ′ (𝑥) − 𝑔′(𝑥 )

Example 6. Find the derivative of 𝑓(𝑥 ) = 4𝑥 5 − 6 Solution: 𝑓 ′(𝑥 ) = 5(4𝑥 )5−1 − 0 𝑓 ′ (𝑥 ) = 20𝑥 4

Apply the constant multiple rule, subtract, then simplify

Product Rule Let f(x) and g(x) be differentiable functions. Then

that is,

𝑑 (𝑓(𝑥)𝑔(𝑥)) 𝑑𝑥

𝑑

= 𝑑𝑥 (𝑓(𝑥)). 𝑔(𝑥) +

𝑑 (𝑔(𝑥)) . 𝑓(𝑥) 𝑑𝑥

ℎ′ (𝑥 ) = 𝑓 ′ (𝑥 )𝑔(𝑥 ) + 𝑔′ (𝑥 )𝑓(𝑥) Example 7. Find the derivative of ℎ(𝑥 ) = (𝑥 2 − 3)( 3𝑥 2 + 4𝑥), find ℎ′ (𝑥). Solution: ℎ′ (𝑥 ) = (2𝑥 2−1 − 0)(3𝑥 2 + 4𝑥 ) + (2(3𝑥)2−1 + 4𝑥 1−1 )(𝑥 2 − 3)

The derivative of the first factor times the second factor plus the derivative of the second factor times the first factor. Simplify using the ℎ′ (𝑥 ) = (2𝑥 )(3𝑥 2 + 4𝑥 ) + (6𝑥 + 4)(𝑥 2 − 3) distributive property

ℎ′ (𝑥 ) = (6𝑥 3 + 8𝑥 2 ) + (6𝑥 3 + 4𝑥 2 − 18𝑥 − 12)

Add and combine like terms

ℎ (𝑥 ) = (𝑥 2 − 3)( 3𝑥 2 + 4𝑥)

Factor out and combine like terms

ℎ′ (𝑥 ) = 12𝑥 3 + 12𝑥 2 − 18𝑥 − 12

To check, simply

ℎ (𝑥 ) = 3𝑥 4 −4𝑥 3 − 9𝑥 2 − 12𝑥 6

ℎ(𝑥 ) = 4(3𝑥)4−1−3(4𝑥)3−1 − 2(9𝑥)2−1 − 1(12𝑥)1−1 Use the power rule ′(

and simplify.

ℎ 𝑥 ) = 12𝑥 + 12𝑥 − 18𝑥 − 12 3

2

Quotient Rule Let f(x) and g(x) be differentiable functions. Then

that is,

𝑑 𝑓(𝑥) ( ) 𝑑𝑥 𝑔(𝑥)

𝑑

𝑑

(𝑓(𝑥)).𝑔(𝑥)− 𝑑𝑥(𝑔(𝑥)).𝑓(𝑥)

= ( 𝑑𝑥

ℎ ′ (𝑥 ) = (

)

(𝑔(𝑥))2

′

𝑓 (𝑥)𝑔(𝑥)−𝑔′(𝑥)𝑓(𝑥)

)

(𝑔(𝑥))2 2

3𝑥 Example 8. Find the derivative of ℎ(𝑥 ) = 2𝑥+1. Solution: Let 𝑓(𝑥 ) = 3𝑥 2 and 𝑔(𝑥 ) = 2𝑥 + 1. Thus, 𝑓 ′ (𝑥 ) = 6𝑥 and 𝑔′ (𝑥 ) = 2.

ℎ′(𝑥) = (

6𝑥(2𝑥 + 1) − 2(3𝑥 2 )

)

(2𝑥 + 1)2

ℎ′(𝑥) = (

12𝑥 2 + 6𝑥 − 6𝑥 2 ) (2𝑥 + 1)2

ℎ′(𝑥) =

The derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function. Then, simplify.

6𝑥2 + 6𝑥 (2𝑥 + 1)2

Derivative of Exponential and Logarithmic Functions The derivative of the exponential function is equal to the value of the function and their corresponding inverse function is called logarithmic functions. 1. If 𝑓(𝑥) = 𝑒 𝑥 , then 𝑓′(𝑥) = 𝑒 𝑥 . In general: 𝑓(𝑥 ) = 𝑒𝑔(𝑥) , then 𝑓′(𝑥) = (𝑔′ (𝑥 ))𝑒𝑔(𝑥) 2. If 𝑓(𝑥) = 𝑎 𝑥 , 𝑎 > 0, 𝑎 ≠ 1, then 𝑓′(𝑥) = 𝑎 𝑥 ln(𝑎) In general: 𝑓(𝑥 ) = 𝑎𝑔(𝑥), 𝑎 > 0, 𝑎 ≠ 1, then 𝑓′(𝑥) = (𝑔′ (𝑥 ))𝑎𝑔(𝑥) ln(𝑎) 1

3. If 𝑓(𝑥) = ln𝑥, then 𝑓′(𝑥) = 𝑥

4. If 𝑓(𝑥) = 𝑙𝑜𝑔𝑎 𝑥, 𝑎 > 0, 𝑎 ≠ 1, then 𝑓′(𝑥) =

1

(ln𝑎)𝑥

Note that exponential function 𝑓(𝑥 ) = 𝑒𝑥 has the special property that is derivative is the function itself. 𝑓 ′(𝑥) = 𝑒𝑥 = 𝑓(𝑥).

e: the base of the natural logarithm, 2.718281828459045…

7

Example 9. Find the derivative of 𝑓(𝑥 ) = 𝑒𝑥 . Solution: 2

𝑓 ′(𝑥) = 𝑒 𝑥

2

Use the power rule then apply the derivative of exponential function #1, then simplify.

𝑓 ′(𝑥) = 2𝑥(𝑒 𝑥 ) 2

Example 10. Find the derivative of 𝑓(𝑥 ) = 𝑒𝑥 Solution:

2+4

𝑓 ′ (𝑥) = 𝑒 𝑥

2 +4

.

(2𝑥)

𝑓 ′ (𝑥) = 2𝑥(𝑒 𝑥

2+4

)

Use the power rule then apply the derivative of exponential function #1, then simplify.

Example 11. Find the derivative of 𝑓(𝑥 ) = 53𝑥 . Solution: 𝑓 ′(𝑥) = 53𝑥 (ln5)

𝑓 ′(𝑥) = 3 ln 5 (53𝑥 )

Use the power rule then apply the derivative of exponential function #2, then simplify.

Example 12. Find the derivative of 𝑓(𝑥 ) = 3𝑥 − 4𝑙𝑜𝑔5 𝑥 . Solution: 𝑓 ′(𝑥) = 3𝑥 ln3 −

4 𝑥 ln 5

Apply the derivative of logarithmic function #4, then simplify.

Example 13. Find the derivative of 𝑓(𝑥 ) = 2𝑒𝑥 + 6𝑥3 ln𝑥 . Solution: 1 𝑓 ′(𝑥) = 2𝑒 𝑥 + 18𝑥 2 ln𝑥 + 6𝑥 3 ( ) 𝑥 𝑓 ′ (𝑥) = 2𝑒 𝑥 + 18𝑥 2 ln𝑥 + 6𝑥 2

Use the power rule then apply the derivative of exponential and logarithmic, then simplify.

Derivatives of Trigonometric Functions The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine. 𝑓′(sin 𝑥) = cos 𝑥

𝑓′(cos 𝑥) = −sin 𝑥

Example 14. Find the derivative of 𝑓(𝑥 ) = 4𝑥3 sin 𝑥 . Solution:

𝑓 ′(𝑥) = 3(4𝑥 3−1 ) sin 𝑥 + cos x (4𝑥3 ) 𝑓 ′ (𝑥) = 12𝑥 2 sin 𝑥 + 4𝑥3 cos 𝑥 8

The derivative of the first factor times the second factor plus the derivative of the second factor times the first factor, then simplify.

Example 15. Find the derivative of 𝑓(𝑥 ) = Solution:

𝑓 ′(𝑥) =

cos 𝑥 3𝑥2

.

(−sin 𝑥)3𝑥 2 − 6𝑥(cos 𝑥) (3𝑥2 )2

𝑓 ′(𝑥) =

−3𝑥 2 sin 𝑥 − 6𝑥 cos 𝑥 9𝑥4

𝑓 ′(𝑥) =

The derivative of the numerator times the denominator minus the derivative of the denominator times numerator, all divided by the square of the denominator. Then, simplify.

−𝑥 sin 𝑥 − 2 cos 𝑥 3𝑥3

The derivatives of the remaining trigonometric functions are as follows: 𝑓 ′ (tan 𝑥 ) = sec 2 𝑥

𝑓 ′(cot 𝑥 ) = − csc 2 𝑥

𝑓 ′ (sec 𝑥 ) = sec 𝑥 tan 𝑥

𝑓′(csc 𝑥) = −csc 𝑥 cot 𝑥

Derivatives of Inverse Trigonometric Functions The six x trigonometric functions above have corresponding inverse functions and have derivatives, which are summarized as follows: ′ 1. If 𝑓(𝑥) = sin −1 𝑥 = arcsinx, − 2 ≤ 𝑓(𝑥) ≤ 2 , then, 𝑓 (𝑥) = ᴫ

ᴫ

2. If 𝑓(𝑥) = cos −1 𝑥 = arccosx, 0 ≤ 𝑓(𝑥) ≤ ᴫ, then, 𝑓′ (𝑥) = √

3. If 𝑓(𝑥) =

4. If 𝑓(𝑥) =

1

√1−𝑥2

−1

1−𝑥2 1 ᴫ ᴫ ′ tan −1 𝑥 = arctanx, − 2 ≤ 𝑓(𝑥) ≤ 2 , then, 𝑓 (𝑥) = 1−𝑥2 −1 ′ cot −1 𝑥 = arccotx, 0 ≤ 𝑓(𝑥) ≤ ᴫ, then, 𝑓 (𝑥) = 1−𝑥2. ᴫ ′ −1

5. If 𝑓(𝑥) = sec

.

. .

𝑥 = arcsecx, 0 ≤ 𝑓(𝑥) ≤ ᴫ, 𝑓(𝑥) ≠ 2 then, 𝑓 (𝑥) =

′ 6. If 𝑓(𝑥) = csc −1 𝑥 = arccscx, − 2 ≤ 𝑓(𝑥) ≤ 2 , 𝑓(𝑥) ≠ 0 then, 𝑓 (𝑥) = ᴫ

ᴫ

Example 16. Find the derivative of 𝑓(𝑥 ) = cos −1 (3𝑥) . Solution: −1 .3 𝑓 ′ (𝑥) = √1 − (3𝑥)2 𝑓 ′(𝑥) =

−3

√1 − 9𝑥2 9

1

.

𝑥√𝑥2 −1 −1

𝑥√𝑥2 −1

Use the formula #2 then multiply the derivative of 3x. Simplify.

Example 17. Find the derivative of 𝑓(𝑥 ) = arctan(𝑥) 3/2 . Solution: 1 3 𝑓 ′(𝑥) = Use the formula #3 then multiply . 𝑥1/2 3/2 2 1 + (𝑥 ) 2 to the derivative of second factor. 𝑓 ′ (𝑥) =

3 1 . 𝑥1/2 3 1+𝑥 2

𝑓 ′(𝑥) =

3 √𝑥 2(1 + 𝑥3 )

Multiply both numerator and denominator. 𝑥1/2is equal to √𝑥. Simplify.

Let Us Practice Multiple Choice. Choose the letter of the best answer. 1. Find the derivative of f(x) = 8. a. 8x b. 0

c. 8 d. -1

2. Which of the following functions is NOT continuous? 𝑥2 −4

a. 𝑓(𝑥 ) = (𝑥 + 4)2 b. 𝑓(𝑥 ) = 𝑚𝑥 + 𝑏

c. 𝑓(𝑥 ) = 𝑥−2 d. 𝑓(𝑥 ) = 1012

3. Which of the following is NOT differentiable? a. 𝑓(𝑥 ) = (𝑥 + 1)3 c. 𝑓(𝑥 ) = 2𝑥 + 3 ( ) b. 𝑓 𝑥 = |𝑥| d. 𝑓(𝑥 ) = 354

4. Find the derivative of 𝑓(𝑥 ) = 7𝑥 4 . a. 𝑓′(𝑥) = 7𝑥3 + 4 b. 𝑓′(𝑥 ) = 21𝑥 3

c. 𝑓′(𝑥 ) = 28𝑥 12 d. 𝑓′(𝑥 ) = 28𝑥 3

5. Find the derivative of f(x) = 2𝑥 3 − 𝑥 2 + 5𝑥 . a. 𝑓 ′ (𝑥) = 6𝑥 2 − 2𝑥 + 5 c. 𝑓 ′(𝑥) = 6𝑥 3 − 2𝑥 + 5 d. 𝑓 ′(𝑥) = 6𝑥 2 − 2𝑥 b. 𝑓 ′(𝑥) = 2𝑥 3 − 2𝑥

10

Let Us Practice More Application on the Derivatives of Exponential and Logarithmic Functions. Problem Solving Given Problem. A Cebu Pacific plane takes off from Davao City International Airport at sea level and its altitude (in feet) at time t (in minutes) is given by ℎ = 3000 ln(𝑡 + 1) Find the rate of climb at time t = 5 min.

https://images.app.goo.gl/PVptgq8ut1WhSwFv7

Let Us Remember True or False. Read and analyze the questions below. If you think the statement is correct write TRUE, if not write FALSE. _______ 1. The derivative of a power function is a function in which the power on x becomes the coefficient of the term and the power on x in the derivative decreases by 1. _______ 2. The derivative of a constant function is 1. _______ 3. The derivative of a product of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function. _______ 4. The derivative of exponential function is the i...

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