Lecture #9: Rules of Differentiation

Definition of the Derivative Function

Recall the definition of the derivative function.

Derivative of Constant Functions

For any real number \(c\), if \(f(x) = c\), then \(f'(x) = 0\).

Derivative of Constant Functions

Proof: If \(f(x) = c\), then: \[ \begin{aligned} f'(x) &= \lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h} \\ &= \lim\limits_{h \to 0} \frac{c - c}{h} \\ &= \lim\limits_{h \to 0} \frac{0}{h} \\ &= 0. \end{aligned} \]

Constant Functions

Examples:

1. Find \(f'(x)\) if \(f(x) = 42\).

2. Find \(g'(x)\) if \(g(x) = e^{110.09} + \log(16.51) - \pi^i\).

The Constant Multiple Rule

For any real number \(k\), if \(f(x)\) is a differentiable function with derivative \(f'(x)\), then \(\dfrac{d}{dx} \left[ k f(x) \right] = k f'(x)\).

The Constant Multiple Rule

Proof: \[ \begin{aligned} \frac{d}{dx} \left[ k f(x) \right] &= \lim\limits_{h \to 0} \frac{k f(x + h) - k f(x)}{h} \\ &= \lim\limits_{h \to 0} \frac{k \left(f(x + h) - f(x)\right)}{h} \\ &= \lim\limits_{h \to 0} k \left(\frac{f(x + h) - f(x)}{h}\right) \\ &= k \left(\lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}\right) \\ &= k f'(x). \end{aligned} \]

The Constant Multiple Rule

Examples:

1. Find \(f'(x)\) if \(f(x) = 3x\).
2. Find \(g'(x)\) if \(g(x) = 4x^2\).

The Sum Rule

If \(f(x)\) and \(g(x)\) are differentiable functions with derivatives \(f'(x)\) and \(g'(x)\) respectively, then \(\dfrac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)\).

The Sum Rule

Proof: \[ \begin{aligned} \frac{d}{dx} \left[ f(x) + g(x) \right] &= \lim\limits_{h \to 0} \frac{\left(f(x+h) + g(x+h) \right) - \left(f(x) + g(x) \right)}{h} \\ &= \lim\limits_{h \to 0} \frac{\left(f(x+h) - f(x) \right) + \left(g(x+h) - g(x) \right)}{h} \\ &= \lim\limits_{h \to 0} \left[ \left( \frac{f(x+h) - f(x) }{h} \right) + \left( \frac{g(x+h) - g(x)}{h} \right) \right] \\ &= \lim\limits_{h \to 0} \left( \frac{f(x+h) - f(x) }{h} \right) + \lim\limits_{h \to 0} \left( \frac{g(x+h) - g(x)}{h} \right) \\ &= f'(x) + g'(x). \end{aligned} \]

The Sum Rule

Examples:

1. Find \(f'(x)\) if \(f(x) = 3x + 4\).
2. Find \(g'(x)\) if \(g(x) = x^2 + 4x + 8\).

The Power Rule

For any nonzero real number \(n\), if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).

The Power Rule

Proof (for positive integer \(n\)): \[ \begin{aligned} \frac{d}{dx} f(x) &= \lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h} \\ &= \lim\limits_{h \to 0} \frac{(x+h)^n - x^n}{h} \\ &= \lim\limits_{h \to 0} \frac{\left( x^n + n x^{n-1} h + \binom{n}{2} x^{n-2} h^2 + \dots + h^n \right) - x^n}{h} \\ &= \lim\limits_{h \to 0} \frac{n x^{n-1} h + \binom{n}{2} x^{n-2} h^2 + \dots + h^n}{h} \\ &= \lim\limits_{h \to 0} \left( n x^{n-1} + \binom{n}{2} x^{n-2} h + \dots + h^{n-1} \right) \\ &= n x^{n-1}. \end{aligned} \]

The Binomial Theorem

Note: The proof there uses the Binomial Theorem: \[ (x + y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k}. \]

The Power Rule

Examples:

1. Find \(f'(x)\) if \(f(x) = 9x^2 + 6x + 3\).
2. Find \(g'(x)\) if \(g(x) = \dfrac{1}{9x}\).
3. Find \(\ell'(x)\) if \(\ell(x) = \sqrt{4x}\).
4. Find \(q'(x)\) if \(q(x) = \dfrac{1}{\sqrt[4]{x}}\).

The Product Rule

If \(f\) and \(g\) are differentiable functions, then their product \(P(x) = (f \cdot g)(x) = f(x) \cdot g(x)\) is also a differentiable function, and \[ P'(x) = f(x) g'(x) + g(x)f'(x). \]

The Product Rule

Examples:

1. Find \(f'(x)\) if \(f(x) = (3x^2 - 4x)(x^4 - 1)\).
2. Find \(g'(x)\) if \(g(x) = 4x^5 (5x^3 - 3\sqrt{x})\).
3. Find \(\ell'(x)\) if \(\ell(x) = \dfrac{1}{x} (2x^3 + 7x^6)\).

The Quotient Rule

If \(f\) and \(g\) are differentiable functions, then their quotient \(Q(x) = \frac{f(x)}{g(x)}\) is also a differentiable function for all \(x\) where \(g(x) \ne 0\), where \[ Q'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{\left(g(x)\right)^2}. \]

The Quotient Rule

Examples:

1. Find \(f'(x)\) if \(f(x) = \dfrac{x^2 + 4}{x^4}\).
2. Find \(g'(x)\) if \(g(x) = \dfrac{2x^3 + 7x^6}{x}\).