Differentiation using chain/product rule

In summary, the conversation is about differentiating an equation involving the exponential function and applying the chain rule. The person is asking for clarification on the properties of the exponential function and how to properly apply the chain rule in this situation.
  • #1
Baartzy89
16
0
Hi,

Just a question on an example in a maths textbook. See attached image for question below.

So, I understand that if you set u=sin(x) and v=e^-cos(x)
f'(x)=u'.v + u.v'

But I'm stuck looking at e^-cos(x), could it also be classified e^(w)?

Also, the second step in differentiating the above equation seems to only differentiate the -cos(x) of the e^-cos(x)...is this because the derivative of e is simply e?
 

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  • #2
Hey Baartzy89 and welcome to the forums.

You have the right idea about the exponential: we have two properties.

The first is that d/dx e^(x) = e^(x) and the chain rule which says d/dx f(g(x)) = g'(x)f'(g(x)), so you need to consider g(x) and the derivative of f(g(x)).
 

1. What is the chain rule in differentiation?

The chain rule in differentiation is a mathematical method that allows one to compute the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

2. How do you use the chain rule in differentiation?

To use the chain rule in differentiation, you first identify a composite function, where one function is nested inside another. Then, you differentiate the outer function and multiply it by the derivative of the inner function. This gives you the derivative of the composite function.

3. What is the product rule in differentiation?

The product rule in differentiation is a method used to find the derivative of a product of two functions. It states that the derivative of a product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.

4. When do you use the product rule in differentiation?

The product rule is used when you need to find the derivative of a product of two functions. This is often the case when dealing with polynomial functions or other functions that can be written as a product of simpler functions.

5. Can you use the chain and product rules together in differentiation?

Yes, the chain and product rules can be used together in differentiation. This is often necessary when dealing with more complex functions that involve both nested functions and products of functions. In these cases, you can apply the chain rule first and then the product rule to find the derivative of the function.

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