How can the power rule be used on composite functions?

[frozonecom]

Homework Statement

Get the first derivative of $f(x)=\sqrt{5x-4}$

Homework Equations

The Attempt at a Solution

The answer given by Wolfram Alpha is $f'(x)=\frac{5}{2\sqrt{5x-4}}$.

The confusion I'm having is what method I should use. Wolfram Alpha suggests that I use the Chain Rule. I want to know if it is possible to use Power Rule since $f(x)=\sqrt{5x-4}$ can be written as $f(x)=(5x-4)^{1/2}$. However, the answer wouldn't be the same as the answer given by Wolfram. I got $f'(x)=\frac{1}{2\sqrt{5x-4}}$

I also tried doing the long way $\frac{f(x+h) - f(x)}{h}$ and couldn't proceed since I can't figure out how to simplify it.

Help would be very much appreciated!

 

[Godwin Kessy]

Can you please illustrate how you went through the power rule?

 

[frozonecom]

Here :)

$f'(x)= (5x-4)^{1/2}$

Let $(y)= 5x-4$

$f'(x)= y^{1/2}$
$f'(x)= (\frac{1}{2})y^{\frac{1}{2} - 1}$
$f'(x)= \frac{y^{-1/2}}{2}$ --- bring y^(-1/2) to make the exponent positive
$f'(x)= \frac{1}{2(y^{1/2})}$
$f'(x)= \frac{1}{2\sqrt{y}}$

Then, since $(y)= 5x-4$, then

$f'(x)= (\frac{1}{2\sqrt{5x-4}})$

However, I don't think that this is mathematically correct. But, this is exactly how my teacher explained it to me when I asked her. Thanks in advance for helping! :)

 

[Mark44]

You need to use the chain rule as well (the chain rule form of the power rule).

You have f(x) = (u)^1/2, with u = 5x - 4, so f'(x) = (1/2)u^-1/2 * du/dx. What you're missing is the du/dx factor.

 

[stonecoldgen]

The thing is that the power rule doesn't work on composite functions. I see that they have already helped you in previous posts but Ill give you a suggestion. Whenever you encounter a function that can be differentiated with the power rule, differentiate it with the chain rule instead and you'll get a better sense of it. That's what I did.

 

[frozonecom]

Thanks for the replies! :)

So in short, you cannot JUST use the Power Rule? You really have to use the chain rule?
And also, is it possible to use the long method? Can someone please show me or at least guide me to solve it?? :)

 

[HallsofIvy]

$$f(x+h)- f(x)= \sqrt{5(x+h)- 4}- \sqrt{5x- 4}$$
"rationalize" the numerator- multiply by
$$\frac{\sqrt{5(x+h)- 4}+ \sqrt{5x-4}}{\sqrt{5(x+h)- 4}+ \sqrt{5x-4}}$$
to get
$$\frac{5(x+h)-4 -(5x-4)}{\sqrt{5(x+h)- 4}+ \sqrt{5x-4}}=\frac{5h}{\sqrt{5(x+h)- 4}+ \sqrt{5x-4}}$$

Can you finish?

 

[Mark44]

The thing is that the power rule doesn't work on composite functions.

But many textbooks present what they call the chain rule form of the power rule.

$\frac{d~u^n}{dx} = n\cdot u^{n - 1} \cdot \frac{du}{dx}$

For example, d/dx(sin³(x)) = 3*sin²(x) * cos(x)

I see that they have already helped you in previous posts but Ill give you a suggestion. Whenever you encounter a function that can be differentiated with the power rule, differentiate it with the chain rule instead and you'll get a better sense of it. That's what I did.