# How do you find the second derivative of y=Asin(Bx)?

Jan 13, 2017

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - A {B}^{2} \sin \left(B x\right)$

#### Explanation:

We use

$\frac{d}{\mathrm{dx}} \sin \left(K x\right) = K \cos \left(K x\right)$ and $\frac{d}{\mathrm{dx}} \cos \left(K x\right) = - K \sin \left(K x\right)$

Differentiating:

$y = A \sin \left(B x\right)$

once gives us:

$\frac{\mathrm{dy}}{\mathrm{dx}} = A \left(B \cos \left(B x\right)\right)$
$\setminus \setminus \setminus \setminus \setminus = A B \cos \left(B x\right)$

Differentiating again we get:

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = A B \left(- B \sin \left(B x\right)\right)$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = - A {B}^{2} \sin \left(B x\right)$