Question

Calc derivatives help

Answer 1

x^4 and 4

Explanation:

The formula for finding the derivative of a function includes the following:

f(a+h) - f(a)

and in this case we have

f(4+h) - f(4) => f(4 + h)^4 - 4^4.

Comparing this to

f(a + h) - f(a), we see that a must be 4 and f(x) must be x^4.

Answer 2

The derivative of f(x) = x^4 at a = 4 is determined through the limit definition. Upon simplification, the result is f'(4) = 64, signifying the instantaneous rate of change at that specific point.

Write down the limit definition:

f'(a) = lim (h → 0) [f(a + h) - f(a)] / h

Substitute the values for f(x) and a:

f'(4) = lim (h → 0) [(4 + h)^4 - 256] / h

Expand the numerator using the binomial expansion:

f'(4) = lim (h → 0) [64h + 48h^2 + 12h^3 + h^4] / h

Simplify by canceling out the common term h in the numerator and denominator:

f'(4) = lim (h → 0) (64 + 48h + 12h^2 + h^3)

Apply the limit by plugging in h = 0:

f'(4) = 64 + 0 + 0 + 0 = 64

Therefore, the derivative of f(x) = x^4 at a = 4 is f'(4) = 64.