Question

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

Answer 1

`2 +5/(2x^(1/2)`

Explanation:

find f(g(x)) = f(`sqrtx + 1 ) = 2(sqrtx + 1 )^2 + sqrtx + 1`

now square out the brackets and collect like terms.

noting that `(sqrtx)^2 = x`

`2(sqrtx + 1 )^2 + sqrtx + 1 = 2( x + 2sqrtx + 1 ) + sqrtx + 1`

`rArr f(g(x)) = 2x + 4sqrtx + 2 + sqrtx + 1 = 2x + 5sqrtx + 3`

now rewrite `sqrtx = x^(1/2)`

then f(g(x)) = `2x +5x^(1/2) + 3`

differentiating to obtain.

f'(g(x)) = `2 + 5 xx1/2 x^(-1/2) = 2 + 5/(2x^(1/2)`