What is dy/dx (derivative) if y=(3 + x^2)^15?

Mar 27, 2015

$\frac{\mathrm{dy}}{\mathrm{dx}} = 30 x {\left(3 + {x}^{2}\right)}^{14}$

Solution

You'll need the power rule: $\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n - 1}$
and the chain rule.
(This combination is sometimes called the generalized power rule.)

$\frac{d}{\mathrm{dx}} \left[{\left(g \left(x\right)\right)}^{n}\right] = n {\left(g \left(x\right)\right)}^{n - 1} \cdot g ' \left(x\right)$

or $\frac{d}{\mathrm{dx}} \left({u}^{n}\right) = n {u}^{n - 1} \frac{\mathrm{du}}{\mathrm{dx}}$

$\frac{d}{\mathrm{dx}} \left({\left(3 + {x}^{2}\right)}^{15}\right) = 15 {\left(3 + {x}^{2}\right)}^{14} \cdot 2 x = 30 x {\left(3 + {x}^{2}\right)}^{14}$