# How do you differentiate  y =cos(3sqrtx+7)  using the chain rule?

Dec 31, 2015

Remembering that chain rule states that $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}}$, we can rename $u = 3 \sqrt{x} + 7$.
Now, let's do it, considering $y = \cos \left(u\right)$ and $u = 3 \sqrt{x} + 7$:
$\frac{\mathrm{dy}}{\mathrm{dx}} = \sin \left(u\right) \cdot \frac{3}{2 \sqrt{x}} = \sin \left(3 \sqrt{x} + 7\right) \cdot \frac{3}{2 \sqrt{x}}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 \sin \left(3 \sqrt{x} + 7\right)}{2 \sqrt{x}}$