How do you find the derivative of $y = cos (x^{2})$ $y=cos(x^2)$ ?

Question

How do you find the derivative of $y = cos (x^{2})$ $y=cos(x^2)$ ?

1 Answer

Impact of this question

93052 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-08-06T21:45:11

We will need to employ the chain rule.

The chain rule states:

$\frac{d}{d x} [f (g (x))] = \frac{d}{d [g (x)]} [f (x)] \cdot \frac{d}{d x} [g (x)]$ $d/dx[f(g(x))] = d/(d[g(x)])[f(x)] * d/dx[g(x)]$

In other words, just treat $x^{2}$ $x^2$ like a whole variable, differentiate the outside function first, then multiply by the derivative of $x^{2}$ $x^2$ .

We know that the derivative of $cos u$ $cosu$ is $- sin u$ $-sin u$ , where $u$ $u$ is anything - in this case it is $x^{2}$ $x^2$ . And the derivative of $x^{2}$ $x^2$ is $2 x$ $2x$ .

(if those identities look unfamiliar to you, I may direct you to this page or this page, which have videos for the derivative of $cos u$ $cosu$ and the power rule, respectively)

Anyhow, by the power rule, we now have:

$\frac{d}{d x} [cos (x^{2})] = - sin (x^{2}) \cdot 2 x$ $d/dx[cos(x^2)] = -sin(x^2) * 2x$

Simplify a bit:

$\frac{d}{d x} [cos (x^{2})] = - 2 x sin (x^{2})$ $d/dx[cos(x^2)] = -2xsin(x^2)$

Science

Math

Humanities

... and beyond

How do you find the derivative of $y = cos (x^{2})$ $y=cos(x^2)$ ?

1 Answer

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you find the derivative of y=cos(x2)y=cos(x^2) ?

1 Answer

Related questions

Impact of this question

How do you find the derivative of $y = cos (x^{2})$ $y=cos(x^2)$ ?