Question

If `f(x)= 2 x^2 + x` and `g(x) = 2e^x + 1`, how do you differentiate `f(g(x))` using the chain rule?

Answer 1

`16e^(2x)+10e^x`

Explanation:

`f(g(x)`

`=f(2e^x+1)=2(2e^x+1)^2+2e^x+1`

`color(white)(xxxxxxxxx)=2(4e^(2x)+4e^x+1)+2e^x+1`

`color(white)(xxxxxxxxx)=8e^(2x)+8e^x+2+2e^x+1`

`color(white)(xxxxxxxxx)=8e^(2x)+10e^x+3`

`rArrd/dx(8e^(2x)+10e^x+3)`

`=8e^(2x)xxd/dx(2x)+10e^xlarrcolor(blue)"chain rule"`

`=16e^(2x)+10e^x`

Answer 2

See below.

Explanation:

`f(x)=2x^2+x` , `g(x)=2e^x+1`

`:.`

`f(g(x))=2(2e^x+1)^2+2e^x+1=8e^(2x)+10e^x+3`

Using the chain rule:

`dy/dx=(dy)/(du)*(du)/dx`

Let `u=e^x`

`8u^2+10u+3`

`dy/(dx)(8u^2)=dy/(du)(8u^2) * (du)/(dx)(u)=16u*e^x=16e^(2x)`

`dy/(dx)(10u)=dy/(du)(10u) * (du)/(dx)(u)=10 * e^x=10e^x`

`:.`

`dy/dx(8x^(2x)+10e^x+3)=16e^(2x)+10e^x`