Math finde the diference

Question

Find the Difference quotient f(x+h)-f(x)/h, where h does not equal zero for the function below

f(x)=8x+6

Answer 1

f(x) = 8x +6

f(x+h) = 8(x+h) +6

f(x+h) - f(x) = 8x+8h + 6 -8x -6 = 8h

[f(x+h)-f(x)]/h = 8h/h = 8

which = the derivative f'(x)

short cut derivative rule is that the derivative of a term = the coefficient

and reduce the variable's exponent by 1.

8x becomes 8x^(1-1) = 8x^0 = 8(1) = 8

while the derivative of a constant is always zero

the quotient difference as h goes to zero is the same as the derivative

graphically, the derivative = the slope of the curve or line

slope of f(x) = 8x +6 is 8

f(x+h)-f(x)/h also = the slope for small h as h approaches zero