Derivative using chain rule

In summary: Just remember to include the x in the first term as well: -2xsin(x^2). Overall, your summary would be: In summary, the chain rule is used to find derivatives of composite functions. For the function u sin(x^2), the first derivative is u * 2x cos(x^2) + sin(x^2) u', and the second derivative is 2ucos(x^2)-4x^2sin(x^2)+4xcos(x^2)u'+sin(x^2)u". It is important to remember to include the x in the first term of the second derivative.
  • #1
goldfronts1
32
0
I am trying to find the first and second derivative using the chain rule of the following:

u sin(x^2)

This is what I have but I don't think it is correct. Can someone pls let me know?

first derivative: u * 2x cos(x^2) + sin(x^2) u'

second derivative:
u * 2( x * -2sin(x^2) + cos(x^2)) + 2xcos(x^2)* u' + sin(x^2)*u" + u'* 2xcos(x^2)

Any help thanks
 
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  • #2
That's pretty close: the first term (for your second derivative) is not correct but I think if you take a second look you will figure it out (you just missed one detail). Otherwise, you can combine like terms to clean it up a bit, but it's still a long expression.
 
  • #3
I don't get it :(
 
  • #4
i am assuming u is an other function right?

Think of it this way

[tex] [(u2x)cos(x^2)]'[/tex] in other words think first of (u2x) as a single function, so you can differentiate this using the product rule for (u2x) and
cos(x^2) right? after that apply again the product rule wherever you end up with (u2x)' and you'll be fine.
 
  • #5
This is what I get

2ucos(x^2)-4x^2sin(x^2)+4xcos(x^2)u'+sin(x^2)u"

Is this correct?
Thanks
 
  • #6
goldfronts1 said:
This is what I get

2ucos(x^2)-4x^2sin(x^2)+4xcos(x^2)u'+sin(x^2)u"

Is this correct?
Thanks


You have this question on another post as well, and it looked fine to me as well there.
Well i take this back, you are short of an x in there.
you should have -2xsin(x^2) somewhere at the beggining, on your original result, at fpost #1.
 
Last edited:
  • #7
goldfronts1 said:
This is what I get

2ucos(x^2)-4x^2sin(x^2)+4xcos(x^2)u'+sin(x^2)u"

Is this correct?
Thanks
This looks fine.
 

1. What is the chain rule?

The chain rule is a rule in calculus that allows us to find the derivative of a composite function. In other words, it allows us to find the derivative of a function within another function.

2. How do you use the chain rule to find a derivative?

To use the chain rule, you first need to identify the inner and outer functions. Then, you take the derivative of the outer function and multiply it by the derivative of the inner function.

3. Why is the chain rule important?

The chain rule is important because it allows us to find the derivative of complex functions, which are often used in real-world applications. It also helps us understand the relationship between different variables in a function.

4. Can you provide an example of using the chain rule to find a derivative?

Sure, let's say we have the function f(x) = (x^2+1)^3. To find the derivative, we first identify the inner function as x^2+1 and the outer function as (x^2+1)^3. Then, we take the derivative of the outer function, which is 3(x^2+1)^2, and multiply it by the derivative of the inner function, which is 2x. So, the derivative of f(x) is 6x(x^2+1)^2.

5. Are there any special cases when using the chain rule?

Yes, there are a few special cases when using the chain rule. One is when the inner function is a constant, in which case the derivative of the inner function is 0 and can be omitted. Another special case is when the outer function is raised to a power, in which case we use the power rule in addition to the chain rule.

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