Derivative of x² x of with respect to e³ x+4} is

Correct Answer is : x+2xlog⁡x3e3x+4{x+2 x x}{3 e³ x+4}}3e3x+4x+2xlogx

Correct Answer is : x+2xlog⁡x3e3x+4{x+2 x x}{3 e³ x+4}}3e3x+4x+2xlogx

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Differential Calculus

Question : 15 of 69

Marks: +1, -0

Derivative of

x^{2} \log x

e^{3 x+4}

Solution:

Concept:

Derivative of one function with respect to another is found by dividing their derivatives with respect to a common variable.

Explanation:

Let

u = x^{2} \log x

and

v = e^{3x+4}

We need

\frac{du}{dv}

Compute

\frac{du}{dx}

using product rule:

\frac{du}{dx} = 2x \log x + x^{2} \cdot \frac{1}{x} = 2x \log x + x = x(2\log x + 1)

Compute

\frac{dv}{dx}

\frac{dv}{dx} = e^{3x+4} \cdot 3 = 3e^{3x+4}

Then

\frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{x(2\log x + 1)}{3e^{3x+4}} = \frac{x + 2x \log x}{3e^{3x+4}}

Answer:

Option C:

\frac{x+2x\log x}{3e^{3x+4}}

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