Question

What is the derivative of `y=tan(x^2)` ?

Answer 1

The derivative of `y=tan(x^2)` is `(dy)/(dx) = (2x)[sec^2(x^2)]`

In solving this problem, we will have to make use of the chain rule, which states that for a function `y = f(g(x))` (that is, `y` is a function which is itself a function of another function), `y'(x) = (g'(x))*f'(g(x))`.

In this case, `f(g(x)) = tan(x^2)`. Our `g(x) = x^2` and our `f(g) = tan(g(x))`. Thus, if our function `f(g)` were `f(x)` we would have `f(x) = tan(x)`.

Via the power rule we know that `d/(dx) (x^2) = 2x`, and via our definitions of the derivatives of trigonometric functions, we know that `f'(x) = sec^2(x)`,or `f'(g) = sec^2(g)`

Thus, by using the chain rule shown above, we obtain:

`y'(x) = (2x)sec^2(x^2)`