Differentiating Other Functions

Exponential Derivatives: The function $f(x) = e^x$ is unique in calculus because it is its own derivative, meaning its rate of change at any point is exactly equal to its value. This property makes the natural exponential function the foundation for modeling continuous growth.

Logarithmic Derivatives: The derivative of the natural logarithm, $\frac{d}{dx}(\ln x) = \frac{1}{x}$, describes how the rate of change of a logarithmic scale decreases as the input value increases. It is important to note that this result is specifically for the natural log (base $e$).

Trigonometric Derivatives: Differentiation of periodic functions like $\sin x$ and $\cos x$ results in other trigonometric functions. For example, the rate of change of a sine wave is a cosine wave, representing the $90^\circ$ phase shift in their rates of change.

Base-a Exponential Growth: When the base is not $e$, a scaling factor is required. The derivative $\frac{d}{dx}(a^x) = a^x \ln a$ accounts for the relative 'steepness' of the growth compared to the natural base.

Linear Transformations in Arguments: Applying a constant multiplier $k$ to the input variable, as in $e^{kx}$, results in the constant being pulled out as a coefficient. This is a specific application of the chain rule principle: $\frac{d}{dx}(e^{kx}) = ke^{kx}$.

Logarithmic Consistency: A unique property occurs with logarithms: $\frac{d}{dx}(\ln kx) = \frac{1}{x}$, regardless of the value of $k$. This happens because $\ln(kx) = \ln k + \ln x$, and since $\ln k$ is a constant, its derivative is zero.

Step 1: Identify Function Type: Determine if the function is exponential ($e^x$ or $a^x$), logarithmic ($\ln x$), or trigonometric. Each has a specific derivative 'template' to follow.

Step 2: Handle Multipliers: Look for constants $k$ inside the function, such as $e^{kx}$ or $\sin(kx)$. Multiply the final derivative by $k$ to account for the inner function's derivative.

Step 3: Combine with Basic Rules: Apply standard sum and constant multiple rules alongside the new transcendental derivatives. For example, to differentiate $3e^{2x} + 5\ln x$, differentiate each term independently and sum them.

Check the Formula Booklet: Many trigonometric derivatives like $\sec x$ and $\cot x$ are provided in the standard formula booklet, but basics like $e^x$ and $\sin x$ are often expected to be memorized.

Verify Logarithmic Arguments: Remember that the derivative of $\ln(kx)$ is always $\frac{1}{x}$. Do not write $\frac{k}{x}$, as this is a common trap that ignores the properties of logarithms.

Watch the Signs: Always double-check the signs for trigonometric derivatives. Sine differentiates to positive cosine, but cosine differentiates to negative sine.

The 'Double Constant' Error: Students often forget the $\ln a$ factor when differentiating $a^x$. For instance, the derivative of $2^x$ is $2^x \ln 2$, not simply $x 2^{x-1}$.

Misinterpreting $e^k$: If $k$ is a constant, $e^k$ is also a constant. Its derivative is $0$, not $e^k$. Only apply exponential rules when the variable $x$ is in the exponent.

Trigonometric Confusion: Mixing up the derivatives of $\tan x$ ($\sec^2 x$) and $\sec x$ ($\sec x \tan x$) is a frequent mistake. Use the mnemonic that 'tangent leads to squares' to keep them straight.