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d/dx (e^(2ln x)) = 2x. A) True B) False

Question:

{eq}\dfrac{\mathrm{d}}{\mathrm{d}x} (e^{2 \ln x} ) = 2x {/eq}

A) True

B) False

Derivative of a Function:

The derivaitve of a function {eq}f {/eq} with respect to {eq}x {/eq} is represented as {eq}\dfrac{\mathrm{d}}{\mathrm{d}x} {/eq}. This represents the rate of change of the function with respect to the variable {eq}x{/eq}.

Answer and Explanation: 1

We are given the statement,

$$\dfrac{\mathrm{d}}{\mathrm{d}x} (e^{2 \ln x} ) = 2x. \\ $$

We have to find whether the given statement is true or false. We do so by differentiating the function {eq}(e^{2 \ln x} ) {/eq} with respect to {eq}x{/eq} and comparing it with the right side of the equation.

$$\begin{align} \dfrac{\mathrm{d}}{\mathrm{d}x} (e^{2 \ln x} ) &=\dfrac{\mathrm{d}}{\mathrm{d}x} (e^{ \ln x^2} ) & \left [ \ \because n\ln (m)= \ln (m^n) \ \right ]\\[0.3cm] &=\dfrac{\mathrm{d}}{\mathrm{d}x} (x^2 ) & \left [ \ \because a^{log_a (x)}=x \ \right ]\\[0.3cm] &=\boxed{2x} & \left [ \text{ Apply the power rule: } \dfrac{\mathrm{d} }{\mathrm{d} x}(x^n)=nx^{n-1} \ \right ] \end{align} $$

Since the obtained value is the same as the right side of the equation, the given statement is true.


Learn more about this topic:

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How to Compute Derivatives

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Chapter 20 / Lesson 1
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This lesson briefly reviews what the derivative of a function is. Then we will look at the limit definition of a derivative, use it to compute derivatives, and see a few shortcuts that result from the limit definition of derivatives.


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