If y = (3x^2+1)^3000, find dy/dx| x=0

[ugu]

y = (3x²+1)³⁰⁰

Let u = 3x²+1

Then y = u³⁰⁰

dy/dx = 300u²⁹⁹*du/dx

du/dx = 6x

So dy/dx = 300(3x²+1)²⁹⁹*6x= ....

that is 3000 not 300

 

[ugu]

I agree that as you gain experience you will use u-substitutions less frequently and that you will use the chain rule without intermediate steps. But

[math]\text {Given } y = (3x^2 + 1)^{3000}, \text { find } \dfrac{dy}{dx} \text { at } x = 0.[/math]
Start by finding dy/dx. If you use a u-substitution, that means

[math]\dfrac{dy}{dx} = \dfrac{dy}{du} * \dfrac{du}{dx}.[/math]
[math]u = 3x^2 + 1 \implies \dfrac{du}{dx} = 6x \text { and } y = u^{3000}.[/math]
[math]y = u^{3000} \implies \dfrac{dy}{du} = 3000u^{2999}.[/math]
[math]\therefore \dfrac{dy}{dx} = \dfrac{dy}{du} * \dfrac{du}{dx} = 3000u^{2999} * 6x = 18000x(3x^2 + 1)^{2900}.[/math]
Now substitute 0 for x. What do you get?

Your method works if you follow it carefully.

but if you have to substitute for x replacing all x existence with 0
6x will be 0; 18000x will be 0: 3x^2 will be 0 and all on the right should end up being 0

 

[Harry_the_cat]

but if you have to substitute for x replacing all x existence with 0
6x will be 0; 18000x will be 0: 3x^2 will be 0 and all on the right should end up being 0

Yes that's correct.