Derivative of cos(y^2)

You have entered [src]

   / 2\
cos\y /

\cos{\left(y^{2} \right)}

d /   / 2\\
--\cos\y //
dy

\frac{d}{d y} \cos{\left(y^{2} \right)}

Transform

Detail solution

Let $u = y^{2}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d y} y^{2}$ :
1. Apply the power rule: $y^{2}$ goes to $2 y$
The result of the chain rule is:

$- 2 y \sin{\left(y^{2} \right)}$

The answer is:

- 2 y \sin{\left(y^{2} \right)}

The first derivative [src]

        / 2\
-2*y*sin\y /

- 2 y \sin{\left(y^{2} \right)}

Simplify

The second derivative [src]

   /   2    / 2\      / 2\\
-2*\2*y *cos\y / + sin\y //

- 2 \cdot \left(2 y^{2} \cos{\left(y^{2} \right)} + \sin{\left(y^{2} \right)}\right)

Simplify

The third derivative [src]

    /       / 2\      2    / 2\\
4*y*\- 3*cos\y / + 2*y *sin\y //

4 y \left(2 y^{2} \sin{\left(y^{2} \right)} - 3 \cos{\left(y^{2} \right)}\right)

Simplify