Find the derivative ofƒ(x) = 1/x5in two different ways:using the Power Rule and using theQuotient Rule

Name: Find the derivative ofƒ(x) = 1/x5in two different ways:using the Power
Uploaded: 2024-03-20T06:47:22.000Z
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Description: Find the derivative ofƒ(x) = 1/x5in two different ways:using the Power Rule and using theQuotient Rule

given: f(x)=1x5 the differentiation of the function f(x)=1x5 by using the power rule is: the…

Answered: Find the derivative ofƒ(x) = 1/x5in two…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter4: Calculating The Derivative

Section4.4: Derivatives Of Exponential Functions

Problem 8E: Find derivatives of the functions defined as follows. y=e-x2

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Question

Find the derivative of
ƒ(x) = 1/x⁵
in two different ways:
using the Power Rule and using the
Quotient Rule

Expert Solution

Step 1

given:

$f (x) = \frac{1}{x^{5}}$

the differentiation of the function $f (x) = \frac{1}{x^{5}}$ by using the power rule is:

the function $f (x) = \frac{1}{x^{5}}$ can be written as:

$f (x) = x^{- 5}$

as we know that the power ule of differentiation is:

$\frac{d (x^{n})}{d x} = n x^{n - 1}$

Step 2

therefore,

$\begin{array}{rcl} \frac{d (f (x))}{d x} & = & \frac{d (x^{- 5})}{d x} \\ = & - 5 x^{- 5 - 1} \\ = & - 5 x^{- 6} \\ = & \frac{- 5}{x^{6}} \end{array}$

the differentiation of the function $f (x) = \frac{1}{x^{5}}$ by using the quotient rule is:

as we know that $\frac{d}{d x} (\frac{f (x)}{g (x)}) = \frac{g (x) \frac{d f (x)}{d x} - f (x) \frac{d g (x)}{d x}}{{(g (x))}^{2}}$

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