How do you find the derivative of #cos(pi*x^2)#?

1 Answer
Nov 25, 2015

The derivative is #- 2 sin(pi x^2) * pi x#.

Explanation:

To build the derivative of #cos(pi * x^2)#, you need to use the chain rule.

If #f(u) = cos(u)# and #u(x) = pi x^2#, the derivative is:

#f'(x) = f'(u(x)) * u'(x)#

So, you need to compute the derivatives of #f(u)# and #u(x)#:

#f(u) = cos(u) color(white)(xxxx) => f'(u) = - sin (u)#
#u(x) = pi x^2 color(white)(xxxxxx) => u'(x) = 2 pi x#

Now we can compute the derivative:

#f'(x) = - sin( pi x^2) * 2 pi x = - 2 sin(pi x^2) * pi x#