How do you find the derivative of f(x)= cos^(4) (5x)?

Nov 1, 2016

Remember that ${\cos}^{4} \left(5 x\right) = {\left(\cos \left(5 x\right)\right)}^{4}$ And apply the chain rule (twice).

Explanation:

$f ' \left(x\right) = 4 {\left(\cos \left(5 x\right)\right)}^{3} \cdot \frac{d}{\mathrm{dx}} \left(\cos \left(5 x\right)\right)$

$= 4 {\left(\cos \left(5 x\right)\right)}^{3} \cdot \left[\sin \left(5 x\right) \cdot \frac{d}{\mathrm{dx}} \left(5 x\right)\right]$

$= 4 {\left(\cos \left(5 x\right)\right)}^{3} \cdot \left[\sin \left(5 x\right) \cdot \left(5\right)\right]$

$= 20 {\cos}^{3} \left(5 x\right) \sin \left(5 x\right)$