Find and simplify the difference quotient (f(x + h) − f(x)) h for the following function. f(x) = 7/ x

Question

still cant figure it out, please please help

Answer 1

This is the basis for the derivative in calculus. We put in x + h for x. So we get 7/(x + h) - 7/x, all divided by h. Take the top part. We need a common denominator, x*(x + h). So to get x(x + h) for the first part we multiply top and bottom by x. That gives 7x/(x*(x + h)). Same for the second one. We multiply top and bottom by (x + h). We get 7(x + h)/(x*(x + h)). OK I know this looks just awful but keep going. Multiply through and watch your signs. On the top you have 7x - 7(x + h). This has the common denominator x*(x + h) and all of that is over h. 7x - 7x = 0, so on the top you have -7h. This gives you     -7h/(x*(x + h)) and since this is all divided by h you can rewrite it as multiplied by 1/h. This makes the h go away. You end up with -7/(x*(x + h)). Why are you doing this? The derivative is the limit as h goes to zero of the function you are solving. When h goes to zero in your result, you get -7/x²  which is the derivative of the function 7/x.