Question

$$f(x)=\ln \left(1+e^{2 x}\right), \text { find } f^{\prime}(0).$$

   $$f(x)=\ln \left(1+e^{2 x}\right), \text { find } f^{\prime}(0).$$

Calculus Early Transcendentals

Calculus Early Transcendentals

James Stewart 7th Edition

Chapter 3, Problem 32

Instant Answer

Solved by Expert Dharmendra Jain

10/16/2020

Step 1

Using the chain rule, the derivative of $f(x)$ is given by: $$f^{\prime}(x)=\frac{1}{1+e^{2x}} \cdot \frac{d}{dx}(1+e^{2x})$$ Show more…

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$$f(x)=\ln \left(1+e^{2 x}\right), \text { find } f^{\prime}(0).$$

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Key Concepts

Chain Rule

The chain rule is essential in differentiation when dealing with composite functions. It allows you to differentiate a composite function by taking the derivative of the outer function evaluated at the inner function and multiplying by the derivative of the inner function.

Differentiation of Logarithmic Functions

When differentiating functions involving natural logarithms, the rule applied is that the derivative of ln(g(x)) is g'(x)/g(x) where g(x) is a differentiable function. This rule is a direct application of logarithmic differentiation.

Differentiation of Exponential Functions

Exponential functions, such as those in the form e^(g(x)), differentiate to e^(g(x)) multiplied by the derivative of the exponent g(x). This property is key when exponentials appear inside other functions, requiring the use of the chain rule.

Evaluation of Derivatives

After finding the general expression for the derivative of a function, substituting specific values allows you to determine the instantaneous rate of change at particular points. This evaluation step is crucial for computing derivatives at precise inputs.

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