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### Course: Calculus 1>Unit 3

Lesson 8: Differentiation using multiple rules

# Applying the chain rule and product rule

Example showing multiple strategies for taking a derivative that involves both the product rule and the chain rule.

## Want to join the conversation?

• After squaring (x^2*sin(x)), wouldn't you have to distribute the 3 before distributing into the other function?
• Distribute the 3 into what is then x^4sin^2(x)? if that is what you are referring to, you don't need to distribute since it's all multiplication.
• Isn't it would be much more simple if the derivative is taken w.r.t (x^2 sin x)?
• That's basically the chain rule. In the end you want the derivative with respect to x, which is why you use d/dx The chain rule is the outside function with respect to the inside function times the inside function with respect to x, ot the next inner function if it was more than just one function inside of another.
• how do you solve for dy/dx or is it the same??
• It's the same. In this video he uses (x^2sin(x))^3 to represent y. Since y is a function of x you can just replace y with its function form where it has x.

so if y=x^2 dy/dx can be rewritten as d(x^2)/dx or d/dx (x^2)
• what should i do in this case f(x)=sin3x^2(5x3-1)^1/3?
• That depends on what you are taking the sine of. If you are taking the sine of the whole expression, use the chain rule, than the product rule. Otherwise, start with the product rule.
• If d/dx[h(x)] = cos³(x) -3xcos²(x)sin(x), would it be proper to simplify with the following steps:
1. divide all three terms by cos²(x), leaving:
cos(x) - 3xsin(x)/cos²(x)
2. Multiply the two remaining terms by cos²(x), leaving:
cos²(x)(cos(x)) - 3xsin(x)
• A question on a question: How on earth does the derivative of y=6x^3*(x^(1/2))*csc(x) get factored the way Sal has it= 3x^(5/2)(7-2xcot(x))csc(x)? It would be helpful if the hints didn't glaze over the most difficult part of the problem (the 500 factoring steps he goes through to give an answer).
(1 vote)
• Part 1 explanation:
1. He uses the product rule to put it into proper form and solves
2. Note that for #1 the first block he solves for the regular derivative. Then for block two, d/dx he had to use the product rule to expand it. (indicated by red font in the parenthesis)
3. Then he combines then using distribution.

Part 2 explanation:
1. Product rule
2. After solving for product rule he then applies the chain rule with the terms that haven't been differentiated yet that need to be reduced based off of the past lessons we have been learning.
3. After solving the chain rule and combining the solved portions you can see they resemble each other.

Note: There are many approaches you can use to solve these problems and this video provides examples of the multiple approaches one can think about to solve these equations.