PLEASE: State the product rule for the derivative of a pair of differentiable functions f and g using your favorite notation. Then use the DEFINITION of the derivative to prove the product rule. Briefly justify your reasoning at each step (Unit 1 Test 1 Question 4)

I get how to state the formula, however, I don't understand, according to the solutions how we prove the derivative.

The understand why we do this step:

(fg)'(x) = lim { f(x + h)g(x + h) − f(x)g(x)}/h
h→ 0

However, I don't understand the step immediately after this according to the solutions.

Question

PLEASE: State the product rule for the derivative of a pair of differentiable functions f and g using your favorite notation. Then use the DEFINITION of the derivative to prove the product rule. Briefly justify your reasoning at each step (Unit 1 Test 1 Question 4)

I get how to state the formula, however, I don't understand, according to the solutions how we prove the derivative.

The understand why we do this step:

(fg)'(x) = lim { f(x + h)g(x + h) − f(x)g(x)}/h 
               h→ 0

However, I don't understand the step immediately after this according to the solutions.

amistre64 · Answer

$$\lim_{h->0}\frac{f(x+h)\ g(x+h)-f(x)\ g(x)}{h}$$
i believe the "trick" here is to find a useful form of "0" that will make the transition smoother.

i forget the "correct" version of it; but lets add and subtract the same value to the top; that way the form changes but the value stays the same.

$$\lim_{h->0}\frac{f(x+h)\ g(x+h)-f(x)\ g(x)+[f(x)g(x+h)-f(x)g(x+h)]}{h}$$
rearrange it to something useful

$$\lim_{h->0}\frac{[f(x+h) g(x+h)-f(x)g(x+h)] +[f(x)g(x+h)-f(x)g(x)]}{h}$$
undistribute the terms as such:

$$\lim_{h->0}\frac{g(x+h) [f(x+h)-f(x)] +f(x)[g(x+h)-g(x)]}{h}$$
split the fraction:
$$\lim_{h->0}\frac{g(x+h) [f(x+h)-f(x)]}{h} +\frac{f(x)[g(x+h)-g(x)]}{h}$$
Reform it to match your usual derivativelike format
$$\lim_{h->0}g(x+h) \frac{[f(x+h)-f(x)]}{h} +f(x)\frac{[g(x+h)-g(x)]}{h}$$
the limit of a sum is the sum of the limits ...
$$\lim_{h->0}g(x+h) \frac{[f(x+h)-f(x)]}{h} +\lim_{h->0}f(x)\frac{[g(x+h)-g(x)]}{h}$$
and if i recall correctly the limit of products is the product of limits also...

$$\lim_{h->0}g(x+h) \lim_{h->0}\frac{[f(x+h)-f(x)]}{h} +\lim_{h->0}f(x)\lim_{h->0}\frac{[g(x+h)-g(x)]}{h}$$
and the rest of it is pretty much definitional as well...

g(x) f'(x) + f(x) g'(x)

anonymous · Answer

Yes, that clears it up, thank you. I don't know why I didn't follow my instinct that there was a "adding 0 trick," but now I see it.

thesmartone · Answer

Poor Amistre64, he used to be 100 back then. .-. https://preetharam.files.wordpress.com/2011/12/amistrelimits1.jpg

thesmartone · Answer

And also this :P https://preetharam.files.wordpress.com/2011/12/amistrelevels1.jpg