Let f ( x ) = x /x + 4 . Find the values of x where f ' ( x ) = 3 . The greater solution of x =, the lesser solution of x=

Question

I used the quotient rule to find the derivative and got x(x+4)-x(x) all divded by (x+4)^2. then i simplified it to 4x/(x+4)^2. what do I do now?

Answer 1

You didn't calculate the numerator part of the chain rule properly. It is f'(x)g(x) - g'(x)f(x) = 1(x+4) - 1(x) = 4

Thus, the derivative is 4/(x+4)² Another way of calculating the derivative is even simpler. x/(x+4) = 1 - 4/(x+4)

The derivative of this is 4/(x+4)² , which matches the previously solved value.

Setting this equal to 3, we have 4/(x+4)² = 3

Multiply (x+4)² on both sides, and we have 3(x+4)² = 4 or (x+4)² = 4/3

x + 4 = ± 2/√3 = ±2√3/3

x = ±2√3/3 - 4

The larger value is x = 2√3/3 - 4

The smaller value is x = -2√3/3 - 4