Differentiating Other Functions

The Natural Exponential: The function $f(x) = e^x$ is unique in calculus because its derivative is identical to the function itself, meaning $\frac{d}{dx}(e^x) = e^x$. This property makes it the fundamental building block for modeling growth where the rate of change is proportional to the current value.

General Exponential Bases: For functions with a base other than $e$, such as $a^x$ (where $a > 0$), the derivative is $\frac{d}{dx}(a^x) = a^x \ln a$. The inclusion of the natural logarithm of the base acts as a scaling factor to account for the difference between the base $a$ and the natural base $e$.

Chain Rule Application: When the exponent is a linear function, such as $e^{kx}$, the derivative becomes $ke^{kx}$. Similarly, for $a^{kx}$, the derivative is $k(\ln a)a^{kx}$, which is derived by treating the function as a composite where the inner function is $kx$.

Natural Logarithm: The derivative of the natural logarithm function, $\ln x$, is $\frac{1}{x}$ for $x > 0$. This result is significant because it provides a functional form for the power $x^{-1}$, which cannot be integrated using the standard power rule.

Logarithms with Constants: A common point of confusion is the derivative of $\ln(kx)$. Using the laws of logarithms, $\ln(kx) = \ln k + \ln x$; since $\ln k$ is a constant, its derivative is zero, meaning $\frac{d}{dx}(\ln(kx)) = \frac{1}{x}$, exactly the same as the derivative of $\ln x$.

Base Change Context: While most advanced calculus focuses on the natural log (base $e$), understanding that the derivative of $\ln x$ is $\frac{1}{x}$ is the prerequisite for differentiating logs of other bases using the change of base formula.

Sine and Cosine: The derivatives of the primary trigonometric functions are cyclical: $\frac{d}{dx}(\sin x) = \cos x$ and $\frac{d}{dx}(\cos x) = -\sin x$. It is vital to remember the negative sign when differentiating the cosine function, as this represents the decreasing slope of the cosine wave in its first quadrant.

The Tangent Function: The derivative of $\tan x$ is $\sec^2 x$. This result can be derived using the quotient rule on $\frac{\sin x}{\cos x}$, leading to the identity $\frac{\cos^2 x + \sin^2 x}{\cos^2 x}$, which simplifies to $\frac{1}{\cos^2 x}$.

Secant and Cosecant: The derivative of $\sec x$ is $\sec x \tan x$, and the derivative of $\csc x$ is $-\csc x \cot x$. These are often derived by rewriting the functions as $(\cos x)^{-1}$ and $(\sin x)^{-1}$ respectively and applying the chain rule.

Cotangent: The derivative of $\cot x$ is $-\csc^2 x$. Similar to the tangent derivative, this result involves a squared reciprocal function, but it carries a negative sign, consistent with the derivatives of other 'co-' functions (cosine and cosecant).

The 'Co-' Rule: A helpful mnemonic for trigonometric derivatives is that every function starting with 'co' (cosine, cosecant, cotangent) has a derivative that is negative.

Linear Arguments: Always distinguish between the derivative of the function itself and the derivative of the argument. For example, in $\sin(kx)$, the $k$ must be multiplied out front due to the chain rule, resulting in $k \cos(kx)$.

Formula Book Proficiency: Most of these results are provided in standard formula booklets; students should focus on recognizing the forms rather than rote memorization, ensuring they can identify when a function requires a specific rule.

Chain Rule Awareness: The most common source of lost marks is forgetting to differentiate the 'inner' function. Always check if the argument of the $\ln$, $e$, or trig function is more than just a simple $x$.

Simplification Before Differentiation: Use log laws to simplify expressions like $\ln(x^2)$ to $2\ln x$ before differentiating. This often turns a complex chain rule problem into a simple scalar multiplication.

Sanity Check for Trig: If you differentiate a trigonometric function and the result doesn't 'look' like a trig identity or a standard result, re-verify your signs and the specific function (e.g., confusing $\sec x$ with $\sec^2 x$).