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Handout Wednesday 10/8
Let’s review the chain rule with a few examples, and then see how it can teach us rules for new functions.
Find the derivatives of the following functions.
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\(\displaystyle f(x) = (\sin(x) + 3^x)^5\)
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\(\displaystyle f(x) = (\sin(x)3^x)^5\)
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\(\displaystyle f(x) = \left(\frac{\sin(x)}{3^x}\right)^5\)
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\(\displaystyle f(x) = \cos(x^2 + 7^{\sqrt[3]{x}})\)
Here is another interesting example: \(f(x) = \frac{1}{\cos(x)}\text{.}\) We have done this already using the quotient rule (to find the derivative of \(\sec(x)\)), but notice we could also write this as \(f(x) = (\cos(x))^{-1}\) and then it is a chain rule problem!
Now for something new. Use the chain rule to find the derivative of \(e^{\ln(x)}\text{.}\) Yeah, I know. We don’t the derivative of \(\ln(x)\text{.}\) So just leave that as \([\ln(x)]'\text{.}\)
But wait! \(e^{\ln(x)} = x\) (why?). So this derivative will be the same as \((x)' = 1\text{.}\) So now solve for \([\ln(x)]'\)
