Lost on derivatives

Question

Let f(x) = 7x²

a) Find: f(x+h) = _____

b) Find: f(x+h)-f(x) = _____

f(x+h)-f(x)

c) Find: ------------- = _____

h

d) Find: f '(x) = ______

Answer 1

Hello Kyle, think about this problem as a substitution problem. Example if you have have y=2x+2 and the problem ask to find f(4), you will substitute 4 for x in that expression y = 2x+ 2 or y= 2(4) +2=10. Now that we review of little bit of algebra ,it is time to complete this problem for you.

a) Find f(x+h)............Just "substitute" x+h for x in the expression f(x) = 7x^2

f(x+h)= 7 (x+h)^2= 7 [(x+h)(x+h)]= 7[x^2 +2xh+h^2]=. 7x^2+14xh+7h^2. answer for part a.

b) Find f(x+h) - f(x)= again substitution ; already found f(x+h) above and f(x)= 7x^2, so I am going to "substitute"

f(x+h)- f(x)= (7x^2+14xh+7h^2)- (7x^2)= 7x^2+14xh+7h^2-7x^2= 14xh+7h^2. answer for part b.

c) Find f(x+h)- f(x)

________ = (14xh+7h^2)/(h) =.

h

14xh. +7h^2

------ ------=. 14x +7h cancelling the h in each term . Usually the definition of derivative includes that

h. h

h approaches zero, therefore your final answer is. f' (x) = 14x + 7(0) = 14x. Final answer.

The 14x represents the slope of y=7X^2.

You can further understand this concept of derivative by graphing your original function y=7x^2,then graph the answer or y=14x to see the slope of the original function. Good luck in the rest of your homework.

Answer 2

f(x+h)=7(x² + 2hx + h²)

f(x+h)-f(x)=14hx+7h²

[f(x+h)-f(x)]/h= 14x + 7h

f'(x)=14x