# How do we differentiate f(x)=sin^4(4x^2-6x+1) using chain rule?

Apr 19, 2017

#### Explanation:

For $y = f \left(x\right) = {\sin}^{4} \left(4 {x}^{2} - 6 x + 1\right)$, we can follow the order,

$f \left(x\right) = g {\left(x\right)}^{4}$, $g \left(x\right) = \sin \left(h \left(x\right)\right)$ and $h \left(x\right) = 4 {x}^{2} - 6 x + 1$

Hence $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{df}}{\mathrm{dx}}$

= (df)/(dg)×(dg)/(dh)×(dh)/(dx)

= 4(g(x)^3)×cos(h(x))×(8x-6)

= $4 {\sin}^{3} \left(4 {x}^{2} - 6 x + 1\right) \cos \left(4 {x}^{2} - 6 x + 1\right) \left(8 x - 6\right)$

= $8 \left(4 x - 3\right) {\sin}^{3} \left(4 {x}^{2} - 6 x + 1\right) \cos \left(4 {x}^{2} - 6 x + 1\right)$