Question

How do you differentiate `f(x) = sec(x^2 + 1)^2`?

Answer 1

`4x (x^2 + 1) * sec(x^2 + 1)^2 * tan(X^2 + 1)^2`

Explanation:

The derivative of the sec of a function is the sec of the function multiplied by the tan of the function multiplied by the derivative of the function. And the derivative of a function to a power is, by the power rule, the power times the function raised to one less than the given power, times the derivative of the function. And the derivative of a polynomial function is the power times the coefficient times the variable raised to one less than the given power.

Answer 2

Here is the "How" using notation.

Explanation:

`f(x) = sec(x^2+1)^2`

`d/dx(secu) = secutanu (du)/dx`, so we get

`f'(x) = sec(x^2+1)^2tan(x^2+1)^2 * d/dx((x^2+1)^2)`

`= sec(x^2+1)^2tan(x^2+1)^2 * 2(x^2+1)d/dx(x^2+1)`

`= sec(x^2+1)^2tan(x^2+1)^2 * 2(x^2+1)(2x)`

`= 4x(x^2+1) sec(x^2+1)^2tan(x^2+1)^2`