Question

How do you differentiate `f(x)=e^tan(x)` using the chain rule?

Answer 1

Multiply the derivative of `e^tanx` by the derivative of `tanx` to get `f'(x)=e^(tanx)sec^2x`.

Explanation:

Differentiating this will require use of the chain rule, which, put plainly, states that the derivative of a composite function (like `e^tanx`) is equal to the derivative of the "inside function" (in this case `tanx`) multiplied by the derivative of the whole function (`e^tanx`).

In math terms, we say the derivative of the composite function `f(g(x))` is `f'(g(x))*g'(x)`.

So, the derivative of `e^tanx` will be the derivative of `e^tanx`, which is just `e^tanx` (the derivative of `e` to the anything is `e` to the anything) times the derivative of `tanx`, which is `sec^2x`. That is to say:
`d/dxe^tanx=e^tanx*(tanx)'=e^tanxsec^2x`