If $f(x)= cos5 x$ and $g(x) = e^(3+4x )$ , how do you differentiate $f(g(x))$ using the chain rule?

Question

1629 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-01-05T21:22:00

Leibniz's notation can come in handy.

$f(x)=cos(5x)$

Let $g(x)=u$ . Then the derivative:

$(f(g(x)))'=(f(u))'=(df(u))/dx=(df(u))/(dx)(du)/(du)=(df(u))/(du)(du)/(dx)=$

$=(dcos(5u))/(du)*(d(e^(3+4x)))/(dx)=$

$=-sin(5u)*(d(5u))/(du)*e^(3+4x)(d(3+4x))/(dx)=$

$=-sin(5u)*5*e^(3+4x)*4=$

$=-20sin(5u)*e^(3+4x)$

If f(x)= cos5 x f(x)= cos5 x and g(x) = e^(3+4x ) g(x) = e^(3+4x ) , how do you differentiate f(g(x)) f(g(x)) using the chain rule?