3.3 Differentiation Rules

Differentiation rules let you find derivatives without going back to the limit definition every time. Once you know these rules, you can quickly differentiate polynomials, rational functions, products, and quotients, which is essential for nearly everything else in calculus.

Differentiation Rules

Basic differentiation rules

These three rules handle the simplest cases and show up constantly inside larger problems.

Constant Rule: The derivative of any constant is 0. A constant doesn't change, so its rate of change is zero.

$\frac{d}{dx}(c) = 0$

where $c$ is any constant (like 5, $-2.7$, or $\pi$).

Constant Multiple Rule: If a function is multiplied by a constant, you can pull that constant out front and just differentiate the function.

$\frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)]$

For example, $\frac{d}{dx}(3x^4) = 3 \cdot \frac{d}{dx}(x^4)$.

Power Rule: This is the rule you'll use most often. Bring the exponent down as a coefficient, then reduce the exponent by 1.

$\frac{d}{dx}(x^n) = nx^{n-1}$

where $n$ is any real number. This works for integers, fractions, and negative exponents alike.

• $\frac{d}{dx}(x^3) = 3x^2$
• $\frac{d}{dx}(x^{-1/2}) = -\frac{1}{2}x^{-3/2}$
• $\frac{d}{dx}(x^1) = 1$ (which confirms the derivative of $x$ is 1)

Sum and difference rules

These rules say you can differentiate term by term. You don't need any special technique when functions are added or subtracted.

Sum Rule:

$\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Example: $\frac{d}{dx}(x^2 + \sin x) = 2x + \cos x$

Difference Rule:

$\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}[f(x)] - \frac{d}{dx}[g(x)]$

Example: $\frac{d}{dx}(e^x - \ln x) = e^x - \frac{1}{x}$

Together with the constant multiple rule and power rule, these let you differentiate any polynomial term by term.

Product rule for derivatives

When two functions are multiplied together, you can't just differentiate each one separately. You need the product rule.

$\frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot g'(x) + g(x) \cdot f'(x)$

A common way to remember this: "first times derivative of second, plus second times derivative of first."

Example: Find $\frac{d}{dx}(x^2 \sin x)$.

1. Let $f(x) = x^2$ and $g(x) = \sin x$
2. $f'(x) = 2x$ and $g'(x) = \cos x$
3. Apply the formula: $x^2 \cos x + \sin x \cdot 2x$

Result: $x^2 \cos x + 2x \sin x$

Use the product rule whenever you see two distinct functions multiplied together, such as a polynomial times a trig function, or an exponential times a polynomial.

Quotient rule for derivatives

When one function is divided by another, use the quotient rule. The formula is a bit longer, so take care with signs.

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{[g(x)]^2}$

A mnemonic: "low d-high minus high d-low, over low squared." Here "high" is the numerator $f(x)$ and "low" is the denominator $g(x)$.

Example: Find $\frac{d}{dx}\left(\frac{x^2}{e^x}\right)$.

1. Let $f(x) = x^2$ (numerator) and $g(x) = e^x$ (denominator)

2. $f'(x) = 2x$ and $g'(x) = e^x$

3. Apply the formula: $\frac{e^x(2x) - x^2(e^x)}{(e^x)^2} = \frac{2xe^x - x^2 e^x}{e^{2x}}$

4. Factor: $\frac{e^x(2x - x^2)}{e^{2x}} = \frac{2x - x^2}{e^x}$

A common mistake is getting the subtraction order wrong in the numerator. It's always denominator times derivative of numerator first, then subtract.

Differentiation of negative exponents

Negative exponents aren't a separate rule. They're just the power rule applied to negative values of $n$.

$\frac{d}{dx}(x^{-n}) = -nx^{-n-1}$

This is useful because it means you can rewrite fractions as negative exponents and avoid the quotient rule entirely for simple cases.

• $\frac{d}{dx}(x^{-3}) = -3x^{-4}$
• $\frac{d}{dx}\left(\frac{1}{\sqrt{x}}\right) = \frac{d}{dx}(x^{-1/2}) = -\frac{1}{2}x^{-3/2}$

Whenever you see something like $\frac{1}{x^n}$, consider rewriting it as $x^{-n}$ and using the power rule. It's often faster than setting up the quotient rule.

Complex function differentiation techniques

For functions with multiple terms and operations, the strategy is to break the problem into pieces, differentiate each piece with the right rule, then reassemble.

1. Identify the overall structure: Is it a sum? A product? A quotient?
2. Break the function into its components
3. Apply the appropriate rule (sum, product, or quotient) to the overall structure
4. Use the power rule, constant multiple rule, etc. on each individual piece
5. Combine and simplify

Polynomials are the most straightforward. Use the sum/difference rule to split into terms, then apply the power rule and constant multiple rule to each term.

Example: $\frac{d}{dx}(3x^4 - 2x^3 + 5x - 1) = 12x^3 - 6x^2 + 5$

Rational functions typically require the quotient rule for the overall structure, then simpler rules for the numerator and denominator.

Example: $\frac{d}{dx}\left(\frac{x^2 + 3x}{2x - 1}\right)$

1. Numerator: $f(x) = x^2 + 3x$, so $f'(x) = 2x + 3$

2. Denominator: $g(x) = 2x - 1$, so $g'(x) = 2$

3. Quotient rule: $\frac{(2x-1)(2x+3) - (x^2+3x)(2)}{(2x-1)^2}$

4. Expand numerator: $(4x^2 + 6x - 2x - 3) - (2x^2 + 6x) = 4x^2 + 4x - 3 - 2x^2 - 6x = 2x^2 - 2x - 3$

5. Result: $\frac{2x^2 - 2x - 3}{(2x-1)^2}$

Implicit differentiation is used when $y$ isn't isolated on one side of the equation (like $x^2 + y^2 = 25$). You differentiate both sides with respect to $x$, treating $y$ as a function of $x$, then solve for $\frac{dy}{dx}$. This technique builds directly on the rules covered here.

Advanced concepts in differentiation

Differentiability and continuity: A function must be continuous at a point to be differentiable there. However, continuity alone doesn't guarantee differentiability. Sharp corners (like in $|x|$ at $x = 0$) and vertical tangent lines are continuous but not differentiable.

Higher-order derivatives are found by differentiating repeatedly. The second derivative $f''(x)$ is the derivative of $f'(x)$, the third derivative $f'''(x)$ is the derivative of $f''(x)$, and so on. These come up when analyzing concavity and acceleration.

For example, if $f(x) = x^4$, then $f'(x) = 4x^3$, $f''(x) = 12x^2$, and $f'''(x) = 24x$.