Question

If \(y = x\cdot 7^x\), then the value of \(\frac{dx}{dy}\) when \(x = 1\) is

• A. \(7(log7+1)\)
• B. \(log7+1\)
• C. \(\frac{1}{7(log7+1)}\)
• D. \(\frac{1}{log7+1}\)

Hint:

Find \(\frac{dy}{dx}\) by the product rule and take the reciprocal.

Answer 1

The Correct Option is C

Step 1: Use logarithmic differentiation
$\log y=\log x+x\log7$, so $\dfrac{y'}{y}=\dfrac1x+\log7$.

Step 2: Evaluate
At $x=1$, $y=7$, so $y'=7(1+\log7)$. The reciprocal is $\dfrac{1}{7(1+\log7)}$, option (C).

Final Answer:
$\frac{dx}{dy}=\frac{1}{7(\log7+1)}$, option (C). \[ \boxed{\dfrac{1}{7(\log7+1)}} \]