Differentiation: Product rule and composite rule

In summary, the conversation revolved around using differentiation to verify two integrals, one involving xsinax and the other involving tanax. The homework statement also included the composite rule for differentiating. The attempted solution involved using the product rule for the first integral and applying the chain rule twice for the second integral. However, the correct solutions were not obtained.
  • #1
imy786
322
0

Homework Statement

Use differentiation to verify that the following integrals are correct (where a is not = 0 is a constant and c is an arbitrary constant

(a) integrate xsinax dx= ( −x/a ) (cosax) +(1/a2) sinax+c

(b) integrate tanax dx=(−1/a) ln(cosax)+c

Homework Equations



Composite rule dy/dx = (dy/du) (du/dx)

The Attempt at a Solution



Im not getting the right solution after diffrenting.

attached are my solutions
 
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  • #2
I don't see an attachment. For the first problem, you also need the product rule. For the second one, you need to use the chain rule twice.

(Edited after Mark44's reply below).
 
Last edited:
  • #3
What both of you are referring to as the "composition" rule, most books that I've seen call the chain rule.
 
  • #4
Hm, for some reason I didn't even realize that I was just repeating what he said in a slightly different way. :rolleyes: Yes, the "chain rule" is what I'd normally call it. I don't think I've seen it called anything else in a book.
 

What is the product rule in differentiation?

The product rule is a rule in calculus used to find the derivative of a product of two functions. It states that the derivative of the product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.

What is the composite rule in differentiation?

The composite rule, also known as the chain rule, is a rule in calculus used to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

How do you use the product rule to find the derivative of a product of two functions?

To use the product rule, you first identify the two functions that are being multiplied together. Then, you take the derivative of each function separately. Finally, you add the two derivatives together to get the derivative of the product of the two functions.

Can you give an example of using the composite rule to find the derivative of a composite function?

Sure, let's say we have the function f(x) = (x^2 + 1)^3. To find the derivative of this function, we can use the composite rule by first identifying the outer function as (x^3) and the inner function as (x^2 + 1). Then, we take the derivative of the outer function which is 3x^2 and evaluate it at the inner function, giving us 3(x^2 + 1)^2. Finally, we multiply this by the derivative of the inner function, which is 2x, to get the final derivative of f(x) as 6x(x^2 + 1)^2.

Why are the product rule and composite rule important in calculus?

The product rule and composite rule are important in calculus because they allow us to find the derivative of more complex functions by breaking them down into simpler functions. These rules are essential for solving problems in areas such as physics, economics, and engineering that involve rates of change and optimization.

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