Question

The first derivative of the function $f\left(x\right)=\sin \left(2x\right)\cos \left(2x\right)\cos \left(3x\right)+{\log }_2{2}^{x+3}$ with respect to $x$ at $x=\pi$ is

• A. $2$
• B. $-1$
• C. $1$
• D. None of these

Answer 1

The correct option is B

$-1$

The explanation for the correct answer.

$f\left(x\right)=\sin \left(2x\right)\cos \left(2x\right)\cos \left(3x\right)+{\log }_2{2}^{x+3}$

$=\frac{1}{2}\sin \left(4x\right)\cos \left(3x\right)+(x+3){\log }_{2}2$

$=\frac{1}{2}\sin \left(4x\right)\cos \left(3x\right)+(x+3)$

Differentiate w.r.t $x$

$f\text{'}\left(x\right)=\frac{1}{2}\left[4\cos \left(4x\right)\cos \left(3x\right)–3\sin \left(3x\right)\sin \left(4x\right)\right]+1$ ( Using the product rule of differentiation )

$f\text{'}\left(\pi \right)=\frac{1}{2}[-4–0]+1$

$=-2+1$

$=-1$

Hence, option(B) is the correct answer.