# How do you differentiate f(x)=(sin^3x^2))^(3/2) using the chain rule?

Dec 3, 2015

$f ' \left(x\right) = 9 x \cos \left({x}^{2}\right) {\sin}^{\frac{7}{2}} \left({x}^{2}\right)$

#### Explanation:

Recall that ${\sin}^{3} {x}^{2} = {\left(\sin {x}^{2}\right)}^{3}$.

Thus, we can simplify: $f \left(x\right) = {\left(\sin {x}^{2}\right)}^{\frac{9}{2}}$

Through the chain rule:

$f ' \left(x\right) = \frac{9}{2} {\left(\sin {x}^{2}\right)}^{\frac{7}{2}} \frac{d}{\mathrm{dx}} \left[\sin {x}^{2}\right]$

Find the derivative:

$\frac{d}{\mathrm{dx}} \left[\sin {x}^{2}\right] = \left(\cos {x}^{2}\right) \frac{d}{\mathrm{dx}} \left[{x}^{2}\right] = 2 x \cos {x}^{2}$

Thus:

$f ' \left(x\right) = 9 x \cos \left({x}^{2}\right) {\sin}^{\frac{7}{2}} \left({x}^{2}\right)$