How do you find the derivative of ${cos}^{2} (3 x)$ $cos^2(3x)$ ?

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How do you find the derivative of ${cos}^{2} (3 x)$ $cos^2(3x)$ ?

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Answer 1 · 2016-10-17T08:55:41

$\frac{d}{d x} {cos}^{2} (3 x) = - 6 sin (3 x) cos (3 x)$ $d/(dx)cos^2(3x)=-6sin(3x)cos(3x)$

Explanation:

Using the chain rule, we can treat $cos (3 x)$ $cos(3x)$ as a variable and differentiate ${cos}^{2} (3 x)$ $cos^2(3x)$ in relation to $cos (3 x)$ $cos(3x)$ .

Chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$ $(dy)/(dx)=(dy)/(du)*(du)/(dx)$

Let $u = cos (3 x)$ $u=cos(3x)$ , then $\frac{d u}{d x} = - 3 sin (3 x)$ $(du)/(dx)=-3sin(3x)$

$\frac{d y}{d u} = \frac{d}{d u} u^{2} \to$ $(dy)/(du)=d/(du)u^2->$ since ${cos}^{2} (3 x) = {(cos (3 x))}^{2} = u^{2}$ $cos^2(3x)=(cos(3x))^2=u^2$

$= 2 u = 2 cos (3 x)$ $=2u=2cos(3x)$

$\frac{d y}{d x} = 2 cos (3 x) \cdot - 3 sin (3 x) = - 6 sin (3 x) cos (3 x)$ $(dy)/(dx)=2cos(3x)*-3sin(3x)=-6sin(3x)cos(3x)$

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How do you find the derivative of ${cos}^{2} (3 x)$ $cos^2(3x)$ ?

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How do you find the derivative of ${cos}^{2} (3 x)$ $cos^2(3x)$ ?