Question

Find the derivative of the function f(t) = (6-t)(6 + t^3)^-1

Answer 1

To find the derivative of f(t) = (6-t)(6 + t^3)^-1, we can use the product rule and the chain rule.

Applying the product rule, we have:

f'(t) = (6 - t)'(6 + t^3)^-1 + (6 - t)(6 + t^3)^-1'

The derivative of (6 - t) with respect to t is -1.

The derivative of (6 + t^3)^-1 can be found using the chain rule. Let u = 6 + t^3. Then, du/dt = d(6 + t^3)/dt = 3t^2.

Using the chain rule, we have:

(6 + t^3)^-1' = -1(6 + t^3)^-2 * (6 + t^3)'

= -1(6 + t^3)^-2 * 3t^2

= -3t^2/(6 + t^3)^2

Now, we can substitute the derivatives back into the equation:

f'(t) = -1 * (6 + t^3)^-1 - (6 - t)(3t^2/(6 + t^3)^2)

Simplifying further:

f'(t) = -(6 + t^3)^-1 - (3t^2(6 - t))/(6 + t^3)^2

Therefore, the derivative of f(t) = (6-t)(6 + t^3)^-1 is f'(t) = -(6 + t^3)^-1 - (3t^2(6 - t))/(6 + t^3)^2.