If y=3² x²+7 x+1} then \;dy/dx is

Correct Answer is : 32x2+7x+1⋅log⁡e3⋅(4x+7)3² x²+7 x+1} · _{e} 3 · (4 x+7)32x2+7x+1⋅loge3⋅(4x+7)

3² x²+7 x+1} · _{e} 3 · (4 x+7)

Correct Answer is : 32x2+7x+1⋅log⁡e3⋅(4x+7)3² x²+7 x+1} · _{e} 3 · (4 x+7)32x2+7x+1⋅loge3⋅(4x+7)

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Differential Calculus

Question : 13 of 69

Marks: +1, -0

y=3^{2 x^{2}+7 x+1}

\;\frac{dy}{dx}

Solution:

Concept:

Differentiate an exponential function with base other than

e

using the chain rule:

\frac{d}{dx} a^{u} = a^{u} \cdot \ln a \cdot \frac{du}{dx}

Explanation:

Given

y = 3^{2x^{2}+7x+1}

Let

u = 2x^{2}+7x+1

. Then

\frac{du}{dx} = 4x+7

Differentiating:

\frac{dy}{dx} = 3^{u} \cdot \ln 3 \cdot \frac{du}{dx}

Substitute back:

\frac{dy}{dx} = 3^{2x^{2}+7x+1} \cdot \ln 3 \cdot (4x+7)

Note:

\ln 3 = \log_{e} 3

Answer:

3^{2x^{2}+7x+1} \cdot \log_{e} 3 \cdot (4x+7)

, which matches option B.

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