Can someone help me with this calculus question?

Question

Let f(x)=x/(x+7). Find the value of x where f'(x)=6. Find greater solution and lesser solution of x. I know you have to find the derivative which is -1/(x+1)^2 but I didn't know where to go from there. Thanks!

Answer 1

Set the derivative of f(x) equal to 6.  Use the quotient rule to derive the function.

 

 

f'(x) = 6

 

(x + 7 - x) / (x + 7)² = 6

 

7 / (x + 7)² = 6

 

 

Multiply both sides of the equation by (x + 7)².

 

7 = 6(x² + 14x + 49)

 

7 = 6x² + 84x + 294

 

 

Subtract 7 on both sides of the equation.

 

0 = 6x² + 84x + 287

 

 

Solve this quadratic equation for x.  You may need to use the quadratic formula.