Derivative using chain rule

In summary, the chain rule is a mathematical rule used in calculus to find the derivative of a composite function. To use the chain rule, you first identify the inner and outer functions of the composite function, and then take the derivative of the outer function and multiply it by the derivative of the inner function. It is important because it allows us to find the derivative of complex functions and can be applied to any composite function. While there are common shortcuts and patterns that can make using the chain rule easier, it is important to understand the basic concept in order to use them effectively.
  • #1
goldfronts1
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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
If u is a function of x and you are differentiating with respect to x, then it looks correct to me.
 

What is the chain rule in calculus?

The chain rule is a mathematical rule used in calculus to find the derivative of a composite function. In simpler terms, it allows us to find the rate of change of one variable with respect to another variable.

How do you use the chain rule to find the derivative?

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

Why is the chain rule important?

The chain rule is important because it allows us to find the derivative of complex functions that are made up of multiple functions. It is a fundamental tool in calculus and is used in many real-world applications such as physics, engineering, and economics.

Can the chain rule be applied to any function?

Yes, the chain rule can be applied to any function, as long as it is a composite function. This means that the function is made up of two or more simpler functions.

Are there any shortcuts or tricks to using the chain rule?

There are some common patterns and shortcuts that can make using the chain rule easier, such as the power rule, product rule, and quotient rule. However, it is important to understand the basic concept of the chain rule and how it is applied in order to use these shortcuts effectively.

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