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

`f'(g(x)) = 2 +5/(2sqrtx)`

Explanation:

Find f(g(x)) `= f(sqrtx + 1 )`

substitute `x = sqrtx + 1 " in f(x) to obtain "`

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

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

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

`= 2x + 4sqrtx + 2 + sqrtx + 1 = 2x + 5sqrtx + 3`

and writing `2x + 5sqrtx + 3" as " 2x + 5x^(1/2) + 3`

now differentiate using`color(blue)" power rule "`

ie `d/dx(ax^n) = nax^(n-1) " term by term "`

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