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If {eq}f(x) = \ln(1 + e^{2x}) {/eq}, find {eq}f'(0) {/eq}.

Question:

If {eq}f(x) = \ln(1 + e^{2x}) {/eq}, find {eq}f'(0) {/eq}.

The Chain Rule of Differentiation:

In calculus, the chain rule is a special formula to find the derivative of a composite function. The chain rule states that the derivative of a composite function {eq}f\left( {g\left( x \right)} \right) {/eq} such as {eq}\dfrac{d}{{dx}}f\left( {g(x)} \right) = f'\left( {g\left( x \right)} \right) \times g'\left( x \right) {/eq}.

The following derivative formula may help in the given problem.

{eq}\begin{align*} \dfrac{d}{{dx}}\ln \left( x \right) &= \dfrac{1}{x}\\ \dfrac{d}{{dx}}{e^x} &= {e^x}\\ \dfrac{d}{{dx}}{e^{ax}} &= a{e^{ax}}\\ \dfrac{d}{{dx}}\left( a \right) &= 0 \end{align*} {/eq}

Answer and Explanation: 1

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Given Data:

  • The given function is: {eq}f\left( x \right) = \ln \left( {1 + {e^{2x}}} \right) {/eq}.


Differentiate the function with respect to...

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The Chain Rule for Partial Derivatives

from

Chapter 14 / Lesson 4
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This lesson defines the chain rule. It goes on to explore the chain rule with partial derivatives and integrals of partial derivatives.


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