Home > Calculus > Derivative Rules for y=cos(x) and y=tan(x)

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

Answer 1

#dy/dx = -2xsinx^2#

Process:

The chain rule must be applied to this problem.

If #y = cosx^2#, then, by the chain rule, the derivative will be equal to the derivative of #cosx^2# with respect to #x^2#, multiplied by the derivative of #x^2# with respect to #x#.

We know the basic identity #d/(dx)[cos x] = -sin x#. And, the power rule gives us #d/(dx) [x^2] = 2x#.

(if those identities look unfamiliar to you, some excellent videos can be located here and here, which explain the identity for #cos x# and the power rule, respectively)

So, the derivative of #cosx^2# will therefore be:

#d/(dx) [cos x^2] = -sinx^2 * d/dx[x^2]#

Which reduces even more to:

#d/dx [cos x^2] = -2xsin x^2#

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Answer 2

Luke Alvarez

To find the derivative of ( y = \cos(x^2) ), you would use the chain rule. The derivative is given by ( y' = -2x \sin(x^2) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.