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

Jan 11, 2016

Answer:

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 ${\left(\sqrt{x}\right)}^{2} = x$

$2 {\left(\sqrt{x} + 1\right)}^{2} + \sqrt{x} + 1 = 2 \left(x + 2 \sqrt{x} + 1\right) + \sqrt{x} + 1$

$\Rightarrow f \left(g \left(x\right)\right) = 2 x + 4 \sqrt{x} + 2 + \sqrt{x} + 1 = 2 x + 5 \sqrt{x} + 3$

now rewrite $\sqrt{x} = {x}^{\frac{1}{2}}$

then f(g(x)) = $2 x + 5 {x}^{\frac{1}{2}} + 3$

differentiating to obtain.

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