How do you find y' of y=tan^2(x^4)?
To find the derivative of y = tan^2(x^4), we will use the chain rule of differentiation.
1. Start by applying the power rule to differentiate the outer function, which is squaring:
(d/dx) [tan^2(x^4)] = 2tan(x^4)
2. Now we need to differentiate the inner function, x^4, using the chain rule.
Let's denote u as x^4. Therefore, du/dx = 4x^3.
3. Apply the chain rule, multiplying the derivative of the outer function with the derivative of the inner function:
(d/dx) [tan^2(x^4)] = 2tan(x^4) * du/dx
Substituting du/dx with its value:
(d/dx) [tan^2(x^4)] = 2tan(x^4) * 4x^3
4. Simplify the expression:
(d/dx) [tan^2(x^4)] = 8x^3 * tan(x^4)
Therefore, the derivative of y = tan^2(x^4) is y' = 8x^3 * tan(x^4).