Question

For F(x) = f(x⁵) and G(x) = (f(x))⁵, given a⁴ = 12, f(a) = 3, f'(a) = 15, and f'(a⁵) = 6, find F'(a) and G'(a). Explain the differentiation rules used and show intermediate computational steps.

Answer 1

F'(a) and G'(a) are calculated using the chain rule and power rule of differentiation, yielding results of F'(a) = 360 and G'(a) = 6075.

Step-by-step explanation:

To find F'(a) where F(x) = f(x⁵), we use the chain rule of differentiation which states that if a function h is the composition of two functions f and g, such that h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x). In our case, this translates to F'(x) = f'(x⁵) * (5x⁴).

Plugging in the values we are given, we can calculate:
F'(a) = f'(a⁵) * (5a⁴) = 6 * (5 * 12) = 6 * 60 = 360.

For G'(a) where G(x) = (f(x))⁵, we again use the chain rule along with the power rule (h(x)^n)' = n * h(x)^(n-1) * h'(x). This gives us G'(x) = 5 * (f(x))⁴ * f'(x). Accordingly,
G'(a) = 5 * (f(a))⁴ * f'(a) = 5 * (3)⁴ * 15 = 5 * 81 * 15 = 6075.

These calculations have used the power rule, chain rule, and given information to compute the derivatives at a particular point.