Differentiate Tan^-1y=x^3?

1 Answer
Sep 2, 2017

#dy/dx=3x^2sec^2(x^3)#

Explanation:

#tan^-1y=x^3#

Method 1 - No simplification

Differentiate as written. Recall that #d/dxtan^-1x=1/(1+x^2)#. Here, differentiating #tan^-1y# with respect to #x# will cause the chain rule to be in effect.

#1/(1+y^2)*dy/dx=3x^2#

Solving for the derivative:

#dy/dx=3x^2(1+y^2)#

Let's write this all in terms of #x#. To do this, we have to solve for #y#. From #tan^-1y=x^3#, note that #y=tan(x^3)#.

#dy/dx=3x^2(1+tan^2(x^3))#

Note that #1+tan^2theta=sec^2theta#:

#dy/dx=3x^2sec^2(x^3)#

Method 2 - Simplification

Recall that #y=tan(x^3)#. We can then differentiate this directly, which is easier. It helps to know that #d/dxtanx=sec^2x#. We will use that here and also use the chain rule.

#y=tan(x^3)#

#dy/dx=sec^2(x^3)*d/dxx^3#

#dy/dx=3x^2sec^2(x^3)#