How do you find the derivative of f(x) = (x^2+1)^3?

Mar 30, 2018

$6 x \cdot {\left({x}^{2} + 1\right)}^{2}$

Explanation:

You use the chain rule and bring down the 3 and find the derivative of ${x}^{2} + 1$. The derivative is $2 x$. So, the answer is $2 x \cdot 3 \cdot {\left({x}^{2} + 1\right)}^{2}$. The final answer is $6 x \cdot {\left({x}^{2} + 1\right)}^{2}$

Mar 30, 2018

$6 x {\left({x}^{2} + 1\right)}^{2}$

Explanation:

We use the chain rule, which states that,

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

Let $u = {x}^{2} + 1 , \therefore \frac{\mathrm{du}}{\mathrm{dx}} = 2 x$.

We also have $y = {u}^{3} , \therefore \frac{\mathrm{dy}}{\mathrm{du}} = 3 {u}^{2}$.

Multiplying together, we get,

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {u}^{2} \cdot 2 x$

$= 6 x {u}^{2}$

Undoing the substitution that $u = {x}^{2} + 1$, we get:

$= 6 x {\left({x}^{2} + 1\right)}^{2}$