Question

Differentiate Tan^-1y=x^3?

Answer 1

`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)`