# If f(x)= sec 4 x  and g(x) = 2 x , how do you differentiate f(g(x))  using the chain rule?

Apr 30, 2016

8 sec8x tan 8x

#### Explanation:

f(g(x)=sec4(2x) =sec 8x

Chain rule in simple words is that if z is a function of t and t is a function of x, the $\frac{\mathrm{dz}}{\mathrm{dx}} = \frac{\mathrm{dz}}{\mathrm{dt}} . \frac{\mathrm{dt}}{\mathrm{dx}}$. Inthis case

let g(x) = t= 2x, so f(g(x)= f(t)= sec4t
Hence $\frac{d \left(\sec 4 t\right)}{\mathrm{dx}} = \frac{\mathrm{df}}{\mathrm{dt}} . \frac{\mathrm{dt}}{\mathrm{dx}} = \frac{d \left(\sec 4 t\right)}{\mathrm{dt}} \frac{d \left(2 x\right)}{\mathrm{dx}}$

$= 4 \sec 4 t \tan 4 t \frac{d \left(2 x\right)}{\mathrm{dx}}$

= 4 sec 4t tan 4t (2)

=8 sec8x tan 8x