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17Calculus Derivatives - Frequently Asked Questions

Single Variable Calculus
Multi-Variable Calculus

This page consists of answers to the following frequently asked questions about derivatives.

Which derivative rule is used first?

Which derivative rule is used first?

This section is best understood after you have learned some of the other, more involved rules, including the product rule, quotient rule and chain rule.

A common question we get a lot when we teach differentiation is, which rule is used first, the product rule, quotient rule or chain rule?

The answer is, there is no rule that tells you which one to use first in all cases. You have to decide depending on the problem. We teach our students to take derivatives starting from the outside and working their way in. Here are two examples that might help. We will show you how to get started and then show the complete solutions on the chain rule page.

Example 1:
Evaluate \( \displaystyle{\frac{d}{dx}\left[ (2x)(x^3 +7 )^{10} \right]} \).
If we start on the outside we use the product rule first where \( f(x) = 2x \) and \( g(x) = (x^3 + 7)^{10} \).
Then we use the chain rule on \( g(x) \). [ complete solution on chain rule page ]

Example 2:
Evaluate \( \displaystyle{\frac{d}{dx}\left[ (2x)(x^3 +7) \right]^5} \).
If we start on the outside again, we need to use the chain rule first and then use the product rule. The first step is \( \displaystyle{ 5 \left[ (2x)(x^3+7) \right]^4 \frac{d}{dx}\left[ (2x)(x^3+7)\right] } \). [ complete solution on chain rule page ]

So you can see that there is no rule telling you which to use first. You need to decide by looking at the configuration of the expression.

By the way, some of the chain rule videos show the instructor starting on the inside and working his way out. We find that more difficult than starting on the outside. We recommend that you try both and see which makes more sense for you, unless your instructor requires you to use a specific technique.

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