Differentiate Tan^-1y=x^3?

1 Answer
Sep 2, 2017

dydx=3x2sec2(x3)

Explanation:

tan1y=x3

Method 1 - No simplification

Differentiate as written. Recall that ddxtan1x=11+x2. Here, differentiating tan1y with respect to x will cause the chain rule to be in effect.

11+y2dydx=3x2

Solving for the derivative:

dydx=3x2(1+y2)

Let's write this all in terms of x. To do this, we have to solve for y. From tan1y=x3, note that y=tan(x3).

dydx=3x2(1+tan2(x3))

Note that 1+tan2θ=sec2θ:

dydx=3x2sec2(x3)

Method 2 - Simplification

Recall that y=tan(x3). We can then differentiate this directly, which is easier. It helps to know that ddxtanx=sec2x. We will use that here and also use the chain rule.

y=tan(x3)

dydx=sec2(x3)ddxx3

dydx=3x2sec2(x3)