Question

If $f\left(x\right)=\underset{2x}{\overset{sinx}{\int }}cos\left(t^3\right)dt$, then $f^{\prime }x$ is equal to

• A. $cos(sinx)cosx-2cos(8x^3)$
• B. $sin(si{n}^{3}x)sin-2sin(8x^3)$
• C. $cos(co{s}^{3}x)cosx-2cos(x^3)$
• D. $cos(si{n}^{3}x)-cos(8x^3)$
• E. $sin(si{n}^{3}x)cosx-2sin(8x^3)$

Answer 1

Answer: (A)

Given, $f(x)={\int }_{2x}^{\sin x}\cos t^{3}dt$
Using Leibnitz's rule
$f^{\prime }(x)=\cos \left({\sin }^{3}x\right)\frac{d}{dx}(\sin x)$
$-\cos (2x{)}^3\frac{d}{dx}(2x)$
$=\cos \left({\sin }^{3}x\right)(\cos x)-\cos 8x^3(2)$