Derivative Rules and Higher Derivatives

Derivative Rules

The Constant Rule

The constant rule states that the derivative of a constant function is always zero. This is because a horizontal line has a slope of zero at every point.

• Definition: If $f(x) = k$ for some constant $k$, then $f'(x) = 0$.

• Example: $f(x) = 5 \implies f'(x) = 0$

The Power Rule

The power rule is used to differentiate functions of the form $x^n$ where $n$ is any real number.

• Definition: If $f(x) = x^n$, then $f'(x) = n x^{n-1}$.

• Examples:

  • $f(x) = x^2 \implies f'(x) = 2x$

  • $g(x) = x^{10} \implies g'(x) = 10x^9$

  • $h(x) = \frac{4}{3}x^{13} \implies h'(x) = \frac{52}{3}x^{12}$

  • $m(x) = x^{1/2} \implies m'(x) = \frac{1}{2}x^{-1/2}$

• Common Mistake: Do not multiply the exponent by the base before applying the rule. For example, $f(x) = 3x^4$ should be differentiated as $f'(x) = 12x^3$, not $3x^4$.

Additivity (Sum and Difference Rule)

The sum and difference rule allows you to differentiate sums and differences of functions term by term.

• Definition: If $h(x) = f(x) + g(x)$, then $h'(x) = f'(x) + g'(x)$. Similarly, if $h(x) = f(x) - g(x)$, then $h'(x) = f'(x) - g'(x)$.

• Example: $h(x) = x^3 + x^{1/2}$ $h'(x) = 3x^2 + \frac{1}{2}x^{-1/2}$

Constant Multiple Rule

The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function.

• Definition: If $g(x) = k \cdot f(x)$ for constant $k$, then $g'(x) = k \cdot f'(x)$.

• Example: $g(x) = 5x^7 \implies g'(x) = 5 \cdot 7x^6 = 35x^6$

Combining Sum, Difference, and Constant Multiple Rules

These rules can be combined to differentiate more complex expressions.

• Example: Let $f(x) = 3x^5 - 4x^2 + x + 1$ $f'(x) = 3 \cdot 5x^4 - 4 \cdot 2x + 1 = 15x^4 - 8x + 1$

The Product Rule

The product rule is used to differentiate the product of two functions.

• Definition: If $h(x) = f(x)g(x)$, then $h'(x) = f(x)g'(x) + f'(x)g(x)$.

• Mnemonic: "First times derivative of the second plus second times derivative of the first" (fg' + f'g).

• Example: Let $h(x) = (x^3 + 1)(3x^2 - 7)$ $f(x) = x^3 + 1$, $f'(x) = 3x^2$ $g(x) = 3x^2 - 7$, $g'(x) = 6x$ $h'(x) = (x^3 + 1) \cdot 6x + 3x^2 \cdot (3x^2 - 7)$

• Note: The constant multiple rule is a special case of the product rule when one function is constant.

The Quotient Rule

The quotient rule is used to differentiate the quotient of two functions.

• Definition: If $h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$.

• Mnemonic: "Low d-high minus high d-low over low squared" (denominator times derivative of numerator minus numerator times derivative of denominator, divided by denominator squared).

• Example: $h(x) = \frac{3x^2}{\sqrt{x}e^{x+1}}$ Let $f(x) = 3x^2$, $f'(x) = 6x$ $g(x) = \sqrt{x}e^{x+1}$, $g'(x) = \frac{1}{2\sqrt{x}}e^{x+1} + \sqrt{x}e^{x+1}$ $h'(x) = \frac{\sqrt{x}e^{x+1} \cdot 6x - 3x^2 \cdot (\frac{1}{2\sqrt{x}}e^{x+1} + \sqrt{x}e^{x+1})}{(\sqrt{x}e^{x+1})^2}$

Derivative of the Exponential Function

The derivative of the natural exponential function is unique in that it is the same as the original function.

• Definition: If $f(x) = e^x$, then $f'(x) = e^x$.

• Example: $f(x) = e^{2x} \implies f'(x) = 2e^{2x}$ (by the chain rule, not covered in this set of notes).

Derivatives of Trigonometric Functions

The derivatives of the basic trigonometric functions are as follows:

• $\frac{d}{dx}(\sin x) = \cos x$

• $\frac{d}{dx}(\cos x) = -\sin x$

• $\frac{d}{dx}(\tan x) = \sec^2 x$

• $\frac{d}{dx}(\cot x) = -\csc^2 x$

• $\frac{d}{dx}(\sec x) = \sec x \tan x$

• $\frac{d}{dx}(\csc x) = -\csc x \cot x$

Function	Derivative
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$

• Example: $\frac{d}{dx}(\tan x) = \sec^2 x$

Second and Higher Derivatives

Definition and Notation

Since the derivative of a function is itself a function, it can be differentiated again. The result is called the second derivative. This process can be repeated to obtain higher-order derivatives.

• Notation:

  • First derivative: $f'(x)$ or $\frac{dy}{dx}$

  • Second derivative: $f''(x)$ or $\frac{d^2y}{dx^2}$

  • n-th derivative: $f^{(n)}(x)$ or $\frac{d^n y}{dx^n}$

  • $f^{(0)}(x)$ means the original function $f(x)$ (no differentiation).

Examples of Higher Derivatives

• Example 1: $f(x) = x^5 - x^3 + 1$

  • $f'(x) = 5x^4 - 3x^2$

  • $f''(x) = 20x^3 - 6x$

  • $f'''(x) = 60x^2 - 6$

  • $f^{(4)}(x) = 120x$

  • $f^{(5)}(x) = 120$

  • $f^{(6)}(x) = 0$ (and all higher derivatives are zero)

• Example 2: $f(x) = x^{1/2}$

  • $f'(x) = \frac{1}{2}x^{-1/2}$

  • $f''(x) = -\frac{1}{4}x^{-3/2}$

  • $f'''(x) = \frac{3}{8}x^{-5/2}$

  • ... (never becomes a constant)

• Example 3: $f(x) = \sin x$

  • $f'(x) = \cos x$

  • $f''(x) = -\sin x$

  • $f'''(x) = -\cos x$

  • $f^{(4)}(x) = \sin x$ (cycle repeats every four derivatives)

Summary Table: Higher Derivatives

Function	First Derivative	Second Derivative	Third Derivative	Fourth Derivative
$x^n$	$n x^{n-1}$	$n(n-1)x^{n-2}$	$n(n-1)(n-2)x^{n-3}$	$n(n-1)(n-2)(n-3)x^{n-4}$
$e^x$	$e^x$	$e^x$	$e^x$	$e^x$
$\sin x$	$\cos x$	$-\sin x$	$-\cos x$	$\sin x$
$\cos x$	$-\sin x$	$-\cos x$	$\sin x$	$\cos x$

Key Points

• Use the appropriate rule for each function type (constant, power, sum, product, quotient, exponential, trigonometric).

• Higher derivatives are found by repeated differentiation.

• Some functions (like polynomials) eventually have zero higher derivatives; others (like $e^x$ or $\sin x$) do not.