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

How do you take the derivative of #cos(tan x)#?

Answer 1

William Abdalla

#-sin(tan(x))sec^2(x)#

Use the chain rule, which states that,

#dy/dx=dy/(du)*(du)/dx#

Let #u=tan(x),:.(du)/dx=sec^2(x)#

Then #y=cos(u),:.dy/(du)=-sin(u)#

Combining, we get:

#dy/dx=-sin(u)*sec^2(x)#

#=-sin(u)sec^2(x)#

Substitute back #u=tan(x)# to get the final answer:

#=-sin(tan(x))sec^2(x)#

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Answer 2

Scarlett Allison

To find the derivative of cos(tan(x)), you can use the chain rule. Let u = tan(x). Then cos(tan(x)) = cos(u). Now, find the derivative of cos(u) with respect to u, and then multiply by the derivative of u with respect to x (which is sec^2(x)) by the chain rule. So, the derivative is -sin(u) * sec^2(x). Substitute back u = tan(x) to get the final result: -sin(tan(x)) * sec^2(x).

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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.