Find the difference quotient f(x+h)-f(x)/h , where h is not equal to zero for the function below:
f(x)=5X^2 +3X
Simplify the answer as much as possible.

Question

Find the difference quotient f(x+h)-f(x)/h , where h is not equal to zero for the function below:
f(x)=5X^2 +3X 
Simplify the answer as much as possible.

anonymous · Answer

bunch of algebra
ready?

anonymous · Answer

oh Yes.  I have been trying to work it out but it is confusing.

anonymous · Answer

ok we have $f(x)=5x^2+3x$ so the first thing we need to compute it 
$$f(x+h)=5(x+h)^2+3(x+h)$$ 
by "compute" i really mean remove the parentheses

anonymous · Answer

make sure to square carefully, and don't forget to distribute the $3$ 
$$f(x+h)=5(x+h)^2-3(x+h)$$
$$f(x+h)=5(x+h)(x+h)+3(x+h)$$
$$f(x+h)=5(x^2+2xh+h^2)+3x+3h$$ and finally 
$$f(x+h)=5x^2+10xh+5h^2+3x+3x$$
so far so good?

anonymous · Answer

typo on the last line, should have been 
$$f(x+h)=5x^2+10xh+5h^2+3x+3h$$

anonymous · Answer

you ok to this step? we are not done yet

anonymous · Answer

So far makes perfect sense

anonymous · Answer

from this we have to subtract $f(x)$ again more algebra, don't forget to distribute the minus sign 
$$f(x+h)-f(x)=5x^2+10xh+5h^2+3x+3h-(5x^2+3x)$$
$$f(x+h)-f(x)=f(x+h)=5x^2+10xh+5h^2+3x+3h-5x^2-3x$$

anonymous · Answer

ok i see

anonymous · Answer

another damn typo 
$$f(x+h)-f(x)=5x^2+10xh+5h^2+3x+3h-5x^2-3x$$

anonymous · Answer

when you subtract, all terms without an $h$ in them cancel (add up to zero) leaving only 
$$f(x+h)-f(x)=10xh+3h-5h^2$$

anonymous · Answer

finally you have to divide everything by $h$ 
since there is an $h$ in each term, divide them one by one making
$$\frac{f(x+h)-f(x)}{h}=10x+3-5h$$

anonymous · Answer

That would be 10xh+3h-5h^2/h?

anonymous · Answer

yes, but when you divide by $h$ divide each term since they all have an $h$ in them, and you will end up with no denominator

anonymous · Answer

$$\frac{10xh+3h-5h^2}{h}=10x+3-5h$$

anonymous · Answer

ok I understand

anonymous · Answer

Thanks so much for your help you explained it very clearly.

anonymous · Answer

@satellite73  shouldn't $$5x^2+10xh+5h^2+3x+3h-5x^2-3x simplify \to 10xh+5h^2+3h$$?

anonymous · Answer

@satellite73

anonymous · Answer

yes, did i not write that?

anonymous · Answer

oh damn, i had a minus sign there for no good reason
you are right

anonymous · Answer

ok. Just making sure.  So my final answer is $$10x+5h+3$$

anonymous · Answer

yes

anonymous · Answer

Thanks again for your guidance.