Question

Use the definition of a derivative to find dy/dx of y=f(x)^2.

Answer 1

I know that by taking the derivative the answer will be y' = 2f(x)f'(x), but how do I get that by using the definition of a derivative?

Answer 2

by the definition, do you mean limit f(x+h)- f(x) / h
?

Answer 3

Yes

Answer 4

Just verifying what y ou have there:

Do you want to find the derivative, with respect to x, of y = [f(x)]^2 OR y=x^2?

Answer 5

y = [f(x)]^2

Answer 6

Great. Be sure to use the Chain Rule as well as the Power Rule. By the way, you're in great hands with phi.

Answer 7

I think you are asking to use
\[ \lim_{h\rightarrow 0} \frac{ g(x+h) - g(x)}{h} \]
where g(x) = \(f^2(x) \)

Answer 8

Yes

Answer 9

we get
\[ \frac{f^2(x+h)-f^2(x)}{h} \\
\frac{(f(x+h)-f(x))\cdot (f(x+h)+f(x)}{h}
\]
is that enough of a hint ?

Answer 10

remember the limit of a product is the product of limits
so you can write that as
\[ \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} \cdot \lim_{h\rightarrow 0} (f(x+h)+f(x)) \]

Answer 11

for the sum part, the limit of a sum is the sum of the limits

Answer 12

Ok, so that would simplify to f'(x) * 2f(x)

Answer 13

yes

Answer 14

Ok, thank you for you help!

Answer 15

Though I am thinking there must be another way to do this, because that trick does not work for e.g. f^3(x) (or at least I don't see how)

Answer 16

Probably