Question

The derivative of the function $(x^3+ln6)\left(\sqrt{x}\right)$ is

• A. $\frac{3}{2x^{3/2}}$
• B. $3x^2\sqrt{x}$
• C. $3x^2\sqrt{x}+\frac{x^3+ln6}{2\sqrt{x}}$
• D. $2x^3\sqrt{x}$

Answer 1

The correct option is C $3x^2\sqrt{x}+\frac{x^3+ln6}{2\sqrt{x}}$

The given function is the product of two different functions i.e., $(x^3+ln6)$ and $\sqrt{x}$.
$\frac{d}{dx}\left[f\right(x).g(x\left)\right]=f\left(x\right).g^{\prime }\left(x\right)+g\left(x\right).f^{\prime }\left(x\right)\therefore \frac{d}{dx}\left[\right(x^3+ln6\left)\right(\sqrt{x}\left)\right]=\frac{d}{dx}(x^3+ln6)\times \left(\sqrt{x}\right)+\frac{d}{dx}\left(\sqrt{x}\right)\times (x^3+ln6)=\left(3x^2\right)\left(\sqrt{x}\right)+(x^3+ln6)\left(\frac{1}{2}{x}^{-1/2}\right)=3x^2\sqrt{x}+\frac{x^3+ln6}{2\sqrt{x}}$