If y=x · 7^{x}, then the value of d x/d y when x=1 is

Correct Answer is : 17(log⁡7+1)1/7( 7+1)7(log7+1)1

Correct Answer is : 17(log⁡7+1)1/7( 7+1)7(log7+1)1

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MHTCET 17 Apr 2026 Shift 1 Paper

Question : 121 of 150

Marks: +1, -0

y=x \cdot 7^{x}

\frac{d x}{d y}

x=1

Solution:

Concept:

Use the reciprocal relationship between derivatives:

\frac{dx}{dy} = \frac{1}{dy/dx}

Explanation:

Given

y = x \cdot 7^x

Differentiate both sides with respect to

x

using the product rule.

\frac{dy}{dx} = x \cdot \frac{d}{dx}(7^x) + 7^x \cdot \frac{d}{dx}(x)

We know

\frac{d}{dx}(7^x) = 7^x \log 7

and

\frac{d}{dx}(x) = 1

So,

\frac{dy}{dx} = x \cdot 7^x \log 7 + 7^x

Factor out

7^x

\frac{dy}{dx} = 7^x (x \log 7 + 1)

x = 1

\frac{dy}{dx} = 7^1 (1 \cdot \log 7 + 1) = 7(\log 7 + 1)

Now,

\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{7(\log 7 + 1)}

Answer:

Option C:

\frac{1}{7(\log 7+1)}

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