Find f^'(x). f(x)=5 x-lnx^5

Name: Problem 19, Chapter 3: Calculus for Business, Economics, Life Sciences, and Social Sciences
Uploaded: 2021-02-22T17:51:37-08:00
Duration: 1 min 38 s
Channel: Gregory Higby
Description: Find f^'(x). f(x)=5 x-lnx^5

step 1 So we're finding the derivative of this function, which is 5x minus natural log of x to the 5th.

Question

Find $f^{\prime}(x)$. $f(x)=5 x-\ln x^{5}$

   Find $f^{\prime}(x)$.
$f(x)=5 x-\ln x^{5}$

Calculus for Business, Economics, Life Sciences, and Social Sciences

Raymond A. Barnett,… 13th Edition

Chapter 3, Problem 19

Instant Answer

Solved by Expert Gregory Higby

02/22/2021

Step 1

The law of logarithms states that $\ln a^{n} = n \ln a$. So, we can rewrite the function as: \[f(x) = 5x - 5\ln x\] Show more…

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Find $f^{\prime}(x)$. $f(x)=5 x-\ln x^{5}$

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Key Concepts

Derivative of a Logarithmic Function

This concept describes how to differentiate functions that involve the natural logarithm. The derivative of ln(u(x)) is given by u'(x)/u(x). This rule is fundamental when dealing with logarithmic expressions in calculus, especially when the argument of the logarithm is itself a function of x.

Logarithmic Identity for Exponents

This concept is about recognizing that logarithmic functions can be simplified using exponent rules. The identity ln(a^b) = b ln(a) allows one to bring the exponent out in front of the logarithm. By applying this identity, the differentiation process becomes simpler and more straightforward.

Derivative of a Linear Function

This concept involves finding the rate of change of a function that is linear. For a function of the form ax, the derivative is simply the constant a. This illustrates one of the most basic differentiation rules in calculus, which states that the slope of a linear function is constant.

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