Differentiating Other Functions

Transcendental Functions: These are functions that 'transcend' algebra, meaning they cannot be expressed as a finite sequence of algebraic operations. Examples include exponential ($e^x$), logarithmic ($\\ln x$), and trigonometric functions ($\\sin x, \\cos x$).

Derivative as Rate of Change: For these functions, the derivative still represents the instantaneous rate of change or the gradient of the tangent to the curve at any point $x$.

Base $e$ vs. Base $a$: In calculus, the natural base $e$ is preferred because the derivative of $e^x$ is simply $e^x$, whereas other bases require a scaling factor of $\\ln a$.

Natural Exponential: The derivative of $e^x$ is unique because it is the only function (aside from zero) that is its own derivative. This property makes it foundational for modeling growth and decay.

General Exponential: For any positive base $a$, the derivative of $a^x$ is $a^x \\ln a$. This occurs because $a^x$ can be rewritten as $e^{x \\ln a}$, and differentiating this composite function introduces the constant $\\ln a$.

Linear Exponents: When the exponent is a linear function $kx$, the derivative of $e^{kx}$ is $ke^{kx}$. This is a specific application of the chain rule where the inner function's derivative $k$ multiplies the original function.

Natural Logarithm: The derivative of $\\ln x$ is $\\frac{1}{x}$. This rule is valid for $x > 0$ and is a critical link between transcendental functions and power functions.

Scaled Logarithms: Interestingly, the derivative of $\\ln(kx)$ is also $\\frac{1}{x}$, not $\\frac{k}{x}$. This is because $\\ln(kx) = \\ln k + \\ln x$, and since $\\ln k$ is a constant, its derivative is zero.

Reciprocal Relationship: The derivative of a logarithm results in a rational function, which is a fundamental property used extensively in integration.

Primary Trig Functions: The derivative of $\\sin x$ is $\\cos x$, and the derivative of $\\cos x$ is $-\\sin x$. Note the sign change for the cosine derivative, which is a common point of error.

Tangent Function: The derivative of $\\tan x$ is $\\sec^2 x$. This can be derived using the quotient rule on $\\frac{\\sin x}{\\cos x}$.

Reciprocal Functions: The derivatives of $\\sec x, \\csc x,$ and $\\cot x$ follow specific patterns: $\\frac{d}{dx}(\\sec x) = \\sec x \\tan x$, $\\frac{d}{dx}(\\csc x) = -\\csc x \\cot x$, and $\\frac{d}{dx}(\\cot x) = -\\csc^2 x$.

The 'Co-' Rule: A helpful mnemonic is that the derivatives of all 'co-' functions ($\\cos, \\csc, \\cot$) are negative.

Radian Mode: Always ensure calculations involving trigonometric derivatives are performed in radians. Standard derivative formulas like $\\frac{d}{dx}(\\sin x) = \\cos x$ are mathematically invalid if $x$ is in degrees.

Formula Booklet Usage: While $\\sin x$ and $\\cos x$ are usually expected to be memorized, more complex derivatives like $\\tan x, \\sec x,$ and $\\cot x$ are often provided in exam formula booklets. Always verify which ones you need to know by heart.

The Chain Rule Shortcut: For any function $f(kx + c)$, the derivative is always $k \\cdot f'(kx + c)$. This applies to $e^{kx}$, $\\sin(kx)$, and $\\cos(kx)$ and saves significant time during exams.

Sanity Check for Signs: Before finalizing a trig derivative, check the 'Co-' rule. If the original function starts with 'Co', the derivative must be negative.

The $\\ln(kx)$ Error: A very common mistake is writing the derivative of $\\ln(3x)$ as $\\frac{3}{x}$. Remember that the constant $k$ always cancels out in the derivative of a natural log of a linear term.

Base Confusion: Students often forget the $\\ln a$ term when differentiating $a^x$. If the base is not $e$, you must multiply by the natural log of that base.

Power Rule Misapplication: Never apply the power rule to an exponential function (e.g., differentiating $2^x$ as $x \\cdot 2^{x-1}$ is incorrect).