What is the derivative of ₁0}(5x²+3) with respect to x?

Correct Answer is : 10xlog⁡10e5x2+3{10x ₁0}e}{5x²+3}5x2+310xlog10e

Correct Answer is : 10xlog⁡10e5x2+3{10x ₁0}e}{5x²+3}5x2+310xlog10e

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UPSC NDA I 2017 Mathematics Solved Paper

Question : 114 of 120

Marks: +1, -0

\log_{10}(5x^{2}+3)

x

Solution:

Concept:

Use the change of base formula for logarithms and the chain rule for differentiation.

Explanation:

Given

f(x) = \log_{10}(5x^{2}+3)

Use

\log_{10}a = \frac{\ln a}{\ln 10}

, so

f(x) = \frac{\ln(5x^{2}+3)}{\ln 10}

Differentiate:

f'(x) = \frac{1}{\ln 10} \cdot \frac{1}{5x^{2}+3} \cdot \frac{d}{dx}(5x^{2}+3)

Compute derivative:

\frac{d}{dx}(5x^{2}+3) = 10x

Thus

f'(x) = \frac{10x}{(5x^{2}+3)\ln 10}

Since

\frac{1}{\ln 10} = \log_{10}e

, we have

f'(x) = \frac{10x \log_{10}e}{5x^{2}+3}

Answer:

\frac{10x \log_{10}e}{5x^{2}+3}

(Option C)

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