Confused with when to use chain rule

In summary, the chain rule is used when differentiating a function that has a more complicated argument than just x. This includes functions such as e^x and sin(x), but also extends to more complicated arguments like e^sqrt(x) or sin(x^3). It is safest to always use the chain rule, as it will still give the correct derivative in simpler cases.
  • #1
sl2382
10
0
Just some general questions as I'm confused with when to use chain rule when not to.

For instance, to find the derivative of e^sqrt(x), the right answer is to use chain rule to get e^sqrtx*the derivative of sqrt(x). BUT, isn't there a formula that: d/dx K^x = In(K)*K^x? K for constant and x for differentiable function. So why I can't use it to get e^sqrt(x)=Ine*e^sqrt(x)? AND, isn't d/dx(e^x) = e^x?? I'm completely confused.. Midterm tomorrow.. really need your help.. Thanks a lot!
 
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  • #2
sl2382 said:
Just some general questions as I'm confused with when to use chain rule when not to.

For instance, to find the derivative of e^sqrt(x), the right answer is to use chain rule to get e^sqrtx*the derivative of sqrt(x). BUT, isn't there a formula that: d/dx K^x = In(K)*K^x?
No, there isn't. There is no function named In. Are you thinking ln (LN) for the natural logarithm?
sl2382 said:
K for constant and x for differentiable function. So why I can't use it to get e^sqrt(x)=Ine*e^sqrt(x)? AND, isn't d/dx(e^x) = e^x?? I'm completely confused.. Midterm tomorrow.. really need your help.. Thanks a lot!

d/dx(ex) = ex, but what about d/dx(ef(x))? For that you need the chain rule.

Let u = f(x).
d/dx(ef(x)) = d/dx(eu) = d/du(eu) * du/dx = eu * du/dx.
 
  • #3
I don't know what you mean by In(K). Do you mean ln(K) (natural logarithm of K)? Assuming that's ln:

BUT, isn't there a formula that: d/dx K^x = In(K)*K^x

Yes, but x there isn't a function, it's the variable you're deriving in relation to! The formula for the derivate function of an exponential is:

[tex]\frac{d}{dx}K^{f(x)} = \frac{df(x)}{dx}ln(K)K^{f(x)}[/tex]

The special case when f(x) = x gives that equation you wrote, because [tex]\frac{df(x)}{dx} = 1[/tex]. But when f(x) isn't x, [tex]\frac{df(x)}{dx}[/tex] won't be 1.

To deduce this equation you need to do the chain rule.
 
  • #4
The short answer is you have to use the chain rule whenever the argument isn't just a simple x. So you don't need it for ex. But you need it for e to anything more complicated than that: [itex]e^{2x},\ e^{\sqrt x},\ e^{-x}[/itex] etc.

It's the same with any function. You don't need it for sin(x) but you do for the sine of anything else:[itex]\sin(2x),\ sin(-x),\ sin(x^3)[/itex]etc.

You could be safe and always use it. For example, if you use it on sin(x) you will just get the derivative as [itex]\cos(x)\cdot 1[/itex]. The extra 1 is correct and doesn't hurt anything.
 
  • #5
Thank you so much for all of your helps! Thanks!
 
  • #6
I thought you technically always use the chain rule anyway.

For instance,
[tex]\frac{d}{dx}y=x^2[/tex]
[tex]\frac{dy}{dx}=2x(x')[/tex]
[tex]\frac{dy}{dx}=2x[/tex]

You are still using the chain rule, it's just that x' is 1 because the function is being differentiated with respect to x.
 

What is the chain rule?

The chain rule is a mathematical rule used to find the derivative of a composite function. It states that the derivative of a composite function is equal to the product of the derivative of the outer function and the derivative of the inner function.

When should I use the chain rule?

The chain rule should be used when you have a function that is composed of two or more functions. In other words, when the input of one function is the output of another function.

How do I use the chain rule?

To use the chain rule, you need to identify the outer function and the inner function. Then, take the derivative of the outer function and multiply it by the derivative of the inner function. You may need to use the chain rule multiple times if there are multiple nested functions.

What are some common mistakes when using the chain rule?

Some common mistakes when using the chain rule include not properly identifying the outer and inner functions, forgetting to multiply by the derivative of the inner function, and making errors in the derivative calculations. It is important to double check your work and practice using the chain rule to avoid these mistakes.

Are there any alternative methods to the chain rule?

There are some alternative methods to the chain rule, such as implicit differentiation or logarithmic differentiation. However, the chain rule is typically the most straightforward and efficient method to find the derivative of a composite function.

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