Question

If $y={\sin }^5\theta +{\cos }^8\theta$. Find $\frac{dy}{d\theta }$.

• A. ${\sin }^4\theta {\cos }^3\theta +\tan \theta$
• B. ${\sin }^4\theta {\cos }^3\theta +{\tan }^5\theta$
• C. ${\sin }^2\theta {\cos }^3\theta +{\tan }^5\theta$
• D. $5\cos \theta {\sin }^4\theta -8\sin \theta {\cos }^7\theta$

Answer 1

The correct option is D $5\cos \theta {\sin }^4\theta -8\sin \theta {\cos }^7\theta$

Given, $y={\sin }^5\theta +{\cos }^8\theta$
So, $\frac{dy}{d\theta }=\frac{d({\sin }^5\theta +{\cos }^8\theta )}{d\theta }$
Using chain rule we have
$\frac{d{\sin }^5\theta }{d\sin \theta }\times \frac{d\sin \theta }{d\theta }+\frac{d{\cos }^8\theta }{d\cos \theta }\times \frac{d\cos \theta }{d\theta }$
$\Rightarrow 5\cos \theta {\sin }^4\theta -8\sin \theta {\cos }^7\theta$