Finding Derivatives: Power Rule vs Chain Rule

In summary, when finding the derivative of a value, you can use the power rule if the equation is in the form f(x) = ax^n, or the chain rule for composite functions. The power rule is essentially the chain rule but simpler and easier to use. It is important to practice and understand these rules through examples. The chain rule works for any composition of two functions, while the power rule is a special case where the outer function is a power. It is recommended to refer to a good calculus textbook for a better understanding of these rules.
  • #1
physicskid
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In finding a derivative of a value, how do you know whether when to use the power rule or the chain rule? can anyone please tell me?
 
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  • #2
Usually any good calc textbook will walk you through the process. It's rather mechanical. If you have a function inside of another function, you differentiate the outside one and multiply it by the inside one. With the power rule, you are still using the chain rule without knowing it. For example, in differentiating the function
f(x) = x^2, you take the derivative of the "outside" (x^2) times the derivative of the "inside" (x) yielding f(x) = 2x*(1) . You multiply by one since the chain rule told you to multiply by the derivative of the inside function. I'm not sure if I answered your question, but you'd be better off taking a peek at a good calc textbook (try Stewarts), and working a few problems until you get it. It's really a skill you need to practice to understand how the rules apply.
 
  • #3
If the equation is in the form f(x)=ax^n, then you can use the power rule. If it is a composite function of some form, you can use the chain rule to keep it simple. f(x)=a(x+3)^n can be expanded out and differentiated with the power rule, but it's much easier to use chain.
Although I believe that the power rule is more a derivation of first principles as opposed to application of the chain rule, you can use chain if you want to...but it's so much easier to use anx^n-1 (Power rule).
 
  • #4
Its wery easy, there's no messing around. The power rule is basically the chain rule, but simpler and for easier derivatives.
Lets say we have a function
[tex] u(x) = kx^n [/tex]

[tex] \frac {d}{dx} u(x) = \frac {d(kx^n)}{dx} [/tex]

[tex] \frac {d}{dx} u(x) = (nk)x^{n-1} [/tex]

Now the chain rule. Let's say we have a function:

[tex] f(x) = (u(x))^n [/tex]

[tex] \frac {d}{dx} f(x) = \frac {d((u(x))^n)}{dx} [/tex]

[tex] \frac {d}{dx} f(x)= n(u(x))^n) \frac {d(u(x))}{dx} [/tex]

its a simple set of rules, the best way to get used to them is to practice different examples. Sorry if my notation at the end is a little funky, the latex notation is hard to work with.
 
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  • #5
the chain rule works on the composition of any two functions at all f(g(x)).

the power rule is the special case where the outer function is a power (g(x))^n,

i.e. here u = g(x) is anything, but f(u) = u^n.
 

1. What is the power rule for finding derivatives?

The power rule is a formula used to find the derivative of a function that contains a variable raised to a power. It states that the derivative of xn is equal to n times xn-1, where n is the exponent of x.

2. How is the power rule different from the chain rule?

The power rule is used to find the derivative of a single term with a variable raised to a power, while the chain rule is used to find the derivative of a composite function. In other words, the power rule is used for simple functions, while the chain rule is used for more complex functions.

3. When should I use the power rule instead of the chain rule?

You should use the power rule when the function you are trying to find the derivative of is a simple polynomial with a single term raised to a power. If the function is more complex, with multiple terms or functions within functions, then the chain rule is necessary.

4. What is an example of using the power rule to find a derivative?

Let's say we have the function f(x) = 3x4. To find the derivative of this function, we would use the power rule and multiply the exponent (4) by the coefficient (3), giving us 12x3. Therefore, the derivative of f(x) is f'(x) = 12x3.

5. Can the power rule and chain rule be used together?

Yes, the power rule and chain rule can be used together to find the derivative of a function that contains both a variable raised to a power and a composite function. In these cases, the chain rule is applied to the composite function, and then the power rule is applied to the remaining term with the variable raised to a power.

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