Chain rule Calculus

[jeison1]

Hi, I am studying for a test, and there are two types of problems I have run into where I keep getting stuck while taking derivatives:

1. f(x)=x^2(x-2)^4

I know this is a chain rule, and possibly also a product rule.

I started with the chain rule and got:

d/dx = 4x^2(x-2)^3*(1)

which becomes:

d/dx = 4x^2(x-2)^3

I know that this isn't right.

When I try this with the product rule instead, I get:

f(x)= 4x^2 d/dx(f(x))= 8x
g(x)= (x-2)^3 d/dx(g(x))= 3(x-2)^2*(1) [had to do chain rule here to get derivative]

Which, when laid out with the product rule, becomes:

8x(x-2)^3 + 4x^3(3(x-2)^3 [did not carry (1) since that equals everything to the left]

My instructions are NOT to simplify this. It just doesn't seem right.


2. This one is going to be really hard to write out:

sqrt(2+sqrt(2)*sqrt(x)))

I don't even know how to start writing my work on this out. I know that square roots are a number to the 1/2 power. I'm not sure where to put all of the 1/2 powers, though.

I also don't have to simplify this.

Any help you guys can give would be greatly appreciated. I don't want answers, I just want to know how to deal with the problem to get myself to the right answers.

 

[HallsofIvy]

Hi, I am studying for a test, and there are two types of problems I have run into where I keep getting stuck while taking derivatives:

1. f(x)=x^2(x-2)^4

I know this is a chain rule, and possibly also a product rule.

I started with the chain rule and got:

d/dx = 4x^2(x-2)^3*(1)

which becomes:

d/dx = 4x^2(x-2)^3

I know that this isn't right.

When I try this with the product rule instead, I get:

f(x)= 4x^2 d/dx(f(x))= 8x
g(x)= (x-2)^3 d/dx(g(x))= 3(x-2)^2*(1) [had to do chain rule here to get derivative]

Which, when laid out with the product rule, becomes:

8x(x-2)^3 + 4x^3(3(x-2)^3 [did not carry (1) since that equals everything to the left]

My instructions are NOT to simplify this. It just doesn't seem right.

Well, even if it say not to simplify, I think they would expect you to write "12" instead of 4(3).

2. This one is going to be really hard to write out:

sqrt(2+sqrt(2)*sqrt(x)))

That is \(\displaystyle (2+ \sqrt{2}x^{1/2})^{1/2}\)
Use the chain rule..

I don't even know how to start writing my work on this out. I know that square roots are a number to the 1/2 power. I'm not sure where to put all of the 1/2 powers, though.

I also don't have to simplify this.

Any help you guys can give would be greatly appreciated. I don't want answers, I just want to know how to deal with the problem to get myself to the right answers.

 

[jeison1]

For the first one, I threw in an extra 3 by accident. It actually looks right, I was just over-complicating things. As for the second problem, you just turned on my light bulb Thank you!!!

 

[lookagain]

Hi, I am studying for a test, and there are two types of problems I have run into where I keep getting stuck while taking derivatives:

1. f(x)=x^2(x-2)^4

I know this is a chain rule, and possibly also a product rule.

[Some text was deleted by me on purpose.]


When I try this with the product rule instead, I get:

f(x)= 4x^2 d/dx(f(x))= 8x
g(x)= (x-2)^3 d/dx(g(x))= 3(x-2)^2*(1) [had to do chain rule here to get derivative]

Which, when laid out with the product rule, becomes:

8x(x-2)^3 + 4x^3(3(x-2)^3 [did not carry (1) since that equals everything to the left]

My instructions are NOT to simplify this. It just doesn't seem right.

jeison1,

that is not how you do the chain rule, and not every part used in your product rule
is correct. The errors are to the extent that I would rather display my solution and
have you compare my steps to your attempt.


f(x) = x^2(x - 2)^4

This is a product of functions. One of the ways of separating them could be

g(x) = x^2 and h(x) = (x - 2)^4.

(I am using "g(x)" and "h(x)" so as to distinguish them from "f(x)" notation.)

The strategy of this problem is that you will use the product rule. In using
the product rule, you will make use of the power rule. In determining
h'(x), the chain rule would involve multiplying by 1. Because that is
a relatively trivial part of it, I won't show that step.

Of the many ways of writing the formula for the product rule, in this instance,

I will use f'(x) = [g'(x)][h(x)] + [h'(x)][g(x)].

g(x) = x^2

g'(x) = 2x


h(x) = (x - 2)^4

h'(x) = 4(x - 2)^3


Then f'(x) = [2x][(x - 2)^4] + [4(x - 2)^3][x^2]


If I were not to follow through and do a simplification by way of factoring,
I would probably at least rearrange the expression as shown in this Latex style:


\(\displaystyle f'(x) \ = \ 2x(x - 2)^4 \ + \ 4x^2(x - 2)^3\)