Question

what is chain rule? and how is it different from product rules? * deriative

Answer 1

\[\left(f\circ g (x)\right)'=f'(g(x))\times g'(x)\]

Answer 2

product rule is applied when you're differentiating a product of 2 or more terms

chain rule is differentiating function whiting another function, e.g. d/dx sin(x^2) = cos(x^2) * 2x

Answer 3

so what about (2x^3)(3+1)
is this product?

Answer 4

if you were to differentiate this, 6x^3 + 2x^3, you wouldn't need to apply product rule since they can be differentiated separately.

but yes you could apply product rule as well, note that you second term would be 0 since it doesn't contain X in it

d/dx (2x^3)(3+1) = 6x^2(3+1) + 6x^3(0) [this is rather trivial, since (3+1) =4 and it's just a constant]

Answer 5

it is really hard to distinguish between product and chain rules. could you solve this ? sqrt(3x^2+8)?
i think its apply to product rule?

Answer 6

to distinguish you need to see what happens to X, first you square it then multiply it by 3 then add 8 and then take the square root. so there is a series of functions within functions here which should tell you that you need to use chain rule

Answer 7

start from the outside first, so apply exponent rule first to the square root, and then differentiate inside of the square root

Answer 8

\[f(x)=(3x^2+8)^(1/2) \]

Answer 9

?

Answer 10

if you differentiate it you will get 1/2(3x^2+8)^-1/2 * 6x

Answer 11

when simplified \[\frac{ 3x }{ \sqrt{3x^2+8} }\]

Answer 12

why 6x become 3x?

Answer 13

because 1/2 * 6x = 3x

Answer 14

I missed that.
thats really cool.

Answer 15

Thank you!