Find all values of c such that f'(c) does not exist

Question

Using the function f(x)= (x+7)^2/5

Find all values of c such that f'(c) does not exist

c=?

Answer 1

The derivative of the function is f'(x) = (2/5)(x + 7)^-3/5, which can be rewritten as:

f'(x) = 2/[5(x + 7)^3/5].

The interpretation of (x + 7)^3/5 is the 5th root of (x + 7) raised to the 3rd power. Since you can take an odd root of any number, it would seem that there are no values of x creating a problem... BUT, you cannot divide by zero. And the denominator will equal zero when x + 7 = 0 or x = -7. So the derivative does not exist when

x = -7 (or c = -7).

Check it out here:

desmos.com/calculator/kxqp1c7rhu

Answer 2

Use the Power Rule.

c > - 7