Using chain rule and product rule together
Chain rule and product rule can be used together on the same derivative
We can tell by now that these derivative rules are very often used together. We’ve seen power rule used together with both product rule and quotient rule, and we’ve seen chain rule used with power rule.
In this lesson, we want to focus on using chain rule with product rule. But these chain rule/product rule problems are going to require power rule, too.
Let’s look at an example of how we might see the chain rule and product rule applied together to differentiate the same function.
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Applying chain rule when we differentiate the product of two polynomial functions
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Examples of using chain rule and product rule together
Example
Use chain rule to find the derivative.
???y=8(6xe^x)^{-4}???
Using substitution, we set ???u=6xe^x??? and use product rule to find that
???u'=(6)(e^x)+(6x)(e^x)???
???u'=6e^x+6xe^x???
Our original equation would then look like
???y=8(u)^{-4}???
and according to power rule, the derivative would be
???y'=-32(u)^{-5}u'???
Back-substituting, we get
???y'=-32(6xe^x)^{-5}(6e^x+6xe^x)???
???y'=-\frac{32(6e^x+6xe^x)}{(6xe^x)^5}???
???y'=-\frac{192e^x(x+1)}{7,776x^5e^{5x}}???
???y'=-\frac{2(x+1)}{81e^{4x}x^5}???
Let’s look at another example of chain rule being used in conjunction with product rule.
We’ve seen power rule used together with both product rule and quotient rule, and we’ve seen chain rule used with power rule.
Example
Use chain rule to find the derivative.
???y=(x^2+1)^7(9x^4)???
In this case, ???u=x^2+1??? and ???u'=2x???.
Then the original equation is
???y=(u)^7(9x^4)???
and according to product rule, the derivative is
???y'=7(u)^6(u')(9x^4)+(u)^7(36x^3)???
Back-substituting for ???u??? and ???u'??? gives
???y'=7(x^2+1)^6(2x)(9x^4)+(x^2+1)^7(36x^3)???
???y'=126x^5(x^2+1)^6+36x^3(x^2+1)^7???
???y'=6x^3(x^2+1)^6\left[21x^2+6(x^2+1)\right]???
???y'=6x^3(x^2+1)^6(21x^2+6x^2+6)???
???y'=6x^3(x^2+1)^6(27x^2+6)???
???y'=18x^3(x^2+1)^6(9x^2+2)???
