13. Chain Rule & Implicit Differentiation

Some of the problems in this tutorial, require the derivative of \(\ln\), the natural logarithm. Either ignore those problems, or use the fact that the derivative of \(\ln\) is: \[ \dfrac{d}{dx}\ln x=\dfrac{1}{x} \] In addition, some of the problems require the derivative of \(x^n\) where \(n\) is not a positive integra. Either ignore those problems, or use the fact that the derivative of \(x^n\) is: \[ \dfrac{d}{dx}x^n=n x^{n-1} \qquad \text{for all x} \]

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c. Tutorial on Chain Rule

Use the chain rule to differentiate the composite function:
\((f\circ g)(x)=\)

1. Identify the outer function: (Write it as a function of \(u\).)
   \(f(u)=\)   \(=\) \( \)

   \(f(u)\) is the outer function.

2. Identify the inner function:
   \(g(x)=\)   \(=\) \( \)

   \(g(x)\) is the inner function.

3. Find the derivative of the outer function: (Write it as a function of \(u\))
   \(f'(u)=\)   \(=\) \( \)

   \(f'(u)\) is the derivative of the outer function \(f(u)\) identified in question 1.

4. Find the derivative of the inner function:
   \(g'(x)=\)   \(=\) \(\)

   \(g'(x)\) is the derivative of the inner function \(g(x)\).

5. Find the derivative of the outer function evaluated at the inner function:
   \(f'(g(x))=\)   \(=\) \(\)

   \(f'(g(x))\) is \(f'(u)\) evaluated at \(u = g(x)\).

6. Find the derivative of the composite function.
   \((f\circ g)'(x)=\)   \(=\) \(\)

   \((f\circ g)'(x) = f'(g(x)) g'(x)\), the product \(f'(g(x))\) and \(g'(x)\).
   In words: "The derivative of a composition is the derivative of the outer function evaluated at the inner function times the derivative of the inner function."

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