Question

If $y=5^x{x}^5$, then $\frac{dy}{dx}$ is equal to

• A. $5^x\left(x^5\log 5-5x^4\right)$
• B. $x^5\log 5-5x^4$
• C. $x^5\log 5+5x^4$
• D. $5^x\left(x^5\log 5+5x^4\right)$

Answer 1

The correct option is D

$5^x\left(x^5\log 5+5x^4\right)$

Explanation for the correct option:

Finding the derivative:

Given that,

$y=5^x{x}^5$

Differentiate the given equation with respect to $x$.

$\begin{array}{rcl}\frac{dy}{dx}& =& x^5·5^x\log 5+5^x\left(5\right)x^4\left[\because \frac{d\left(\mathrm{uv}\right)}{\mathrm{dx}}=v\frac{\mathrm{du}}{\mathrm{dx}}+u\frac{\mathrm{dv}}{\mathrm{dx}},\frac{d\left(a^x\right)}{\mathrm{dx}}=\mathrm{loga}·a^x\right]\\ & =& 5^x\left(x^5\log 5+5x^4\right)\end{array}$

Hence, the correct option is D.