1.4: Differentiating a Combination of Functions

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The Sum or Difference Rule

The derivative of the sum of \(f(x)\) and \(g(x)\) is \[(f+g)'=f'+g'.\nonumber\]

Similarly, the derivative of the difference is \[(f-g)'=f'-g'.\nonumber\]

The Product Rule

The derivative of the product of \(f(x)\) and \(g(x)\) is \[(fg)'=f'g+fg',\nonumber\] and should be memorized as “the derivative of the first times the second plus the first times the derivative of the second.”

The Quotient Rule

The derivative of the quotient of \(f(x)\) and \(g(x)\) is \[\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2},\nonumber\] and should be memorized as “the derivative of the top times the bottom minus the top times the derivative of the bottom over the bottom squared.”

The Chain Rule

The derivative of the composition of \(f(x)\) and \(g(x)\) is \[\left(f(g(x))\right)'=f'(g(x))\cdot g'(x),\nonumber\] and should be memorized as “the derivative of the outside times the derivative of the inside."