Derivatives of common functions

Differentiate powers, exponentials, logarithms, and trigonometric functions with physical use.

What it is

Derivatives of common functions are a set of fixed rules that give the derivative of function types that constantly appear: powers (x^n), exponentials (e^x, a^x), logarithms (ln x), and trigonometric functions (sin x, cos x, tan x). Instead of recalculating the limit defining the derivative every time, these rules are memorized and combined with the sum, product, quotient, and composition (chain rule) rules to differentiate more complex expressions.

These rules are tools: each one converts a function of a specific type into another function describing its rate of change, without needing to repeat the limit process every time that functional form appears.

Why it appears in physics

It appears because the functions describing physical phenomena are almost always combinations of these basic types: oscillatory motion is described with sines and cosines, radioactive decay and capacitor charging or discharging are described with exponentials, sound intensity in decibels uses logarithms, and uniformly accelerated motion uses powers of t.

Without these memorized rules, every calculation of velocity, acceleration, current, or decay rate would require recalculating a limit from scratch, impractical for everyday work in physics. The common differentiation rules are the "basic vocabulary" that allows quick progress in real physics problems.

Visual or geometric idea

The visual idea is this: each differentiation rule captures the characteristic way that type of function changes. A power x^n changes faster the larger x is (for n>1), reflected in its derivative nx^(n-1) growing with x. An exponential e^x has the unique property that its rate of change equals its own value, seen in that its slope at each point is proportional to the height of the curve at that point.

Trigonometric functions oscillate, and their derivatives also oscillate but shifted in phase: the slope of sin(x) is maximal where the function crosses zero, and zero where the function reaches its maxima and minima, a pattern visible in any graph of overlaid sine and cosine.

Minimal symbolic language

The basic rules are: d/dx(x^n) = nx^(n-1); d/dx(e^x) = e^x; d/dx(a^x) = a^x ln(a); d/dx(ln x) = 1/x; d/dx(sin x) = cos x; d/dx(cos x) = -sin x. To combine functions: the product rule is d/dx(fg) = f'g + fg', the quotient rule is d/dx(f/g) = (f'g - fg')/g², and the chain rule for a composition f(g(x)) is f'(g(x))·g'(x).

These rules are applied from the outside in when there is composition: first identify the outermost function, differentiate it treating the inner argument as a unit, and then multiply by the derivative of that inner argument.

What the result means

The result of applying these rules is a new function describing the instantaneous rate of change of the original function, for any value of the variable within the domain where the rule applies. The physical meaning depends on the context: if the original function describes position, the result is velocity; if it describes charge, it is current; if it describes oscillating amplitude, the result describes how that amplitude changes at each instant.

A result involving negative exponentials (like -k e^(-kt)) usually describes decay, while one with alternating sines and cosines usually describes oscillation phase-shifted relative to the original function.

When not to use it

Do not apply these rules mechanically without verifying the function has exactly the form for which the rule was defined: for example, the rule d/dx(x^n) = nx^(n-1) requires the exponent to be constant; if the exponent also depends on x (as in x^x), a different technique is needed (logarithmic differentiation).

Also do not forget that the trigonometric functions in the rules above use radians, not degrees: if the argument is in degrees, the derivatives need an additional conversion factor that is frequently omitted by mistake.

Operational summary

Identify the type of function (power, exponential, logarithm, trigonometric, or a combination of them), apply the corresponding basic rule to each part, combine the results using the sum, product, quotient, or chain rule as appropriate, and check that the result makes dimensional and sign sense in the physical context of the problem.

Procedure

Before calculating

Identify the type or types of function present in the expression: powers, exponentials, logarithms, trigonometric functions, or combinations of these through sum, product, quotient, or composition.

Check whether trigonometric function arguments are in radians, and whether there are exponents depending on the variable (which would require an additional technique instead of the simple power rule).

Step 1: identify the mathematical object

Break the expression down into its simplest parts, identifying which operation connects them (sum, subtraction, product, quotient, or composition). If there is a function inside another (for example, sin(3t) or e^(-kt)), identify which is the outer function and which is the inner one.

Explicitly write the derivative of each simple part using the basic rules, before combining them.

Step 2: choose the operation

If the parts are added or subtracted, differentiate each term separately and add the results. If they are multiplied, apply the product rule. If they are divided, apply the quotient rule. If there is composition, apply the chain rule by multiplying the derivative of the outer function (evaluated at the inner one) by the derivative of the inner function.

Combine several rules in the correct order if the expression requires them simultaneously, following the structure of the expression from outside in.

Step 3: preserve units and signs

Check that the sign of each basic derivative is applied correctly: remember especially that the derivative of cos x is -sin x, with a negative sign, a common sign error.

Confirm that the units of the final result are those of the original function divided by the units of the variable, even when the expression combines several types of function.

Step 4: calculate

Apply the rules step by step, without mentally combining several steps at once if the expression is complex; explicitly write the derivative of the outer function evaluated at the inner one, and then multiply by the derivative of the inner one.

Simplify the final result only after having applied all the necessary rules, not before, to avoid premature simplifications that hide an error.

Step 5: interpret the result

Interpret the form of the result in the physical context: a result with a negative exponential usually indicates decay, a result with alternating sines and cosines usually indicates oscillation, and a polynomial result usually indicates a change with acceleration or deceleration depending on the degree of the polynomial.

Check that the result is consistent with the expected behavior of the original function at known points (for example, where the original function has a maximum, the derivative must be zero).

Self-check checklist

Check that you correctly identified the type of each component function, that you applied the chain rule when there was composition, that the signs of the trigonometric and exponential derivatives are correct, and that the final result makes physical sense in the context of the problem.

Guided example

Physical situation

The charge on a capacitor during discharge follows Q(t) = Q0 e^(-t/RC). The task is to calculate the discharge current I(t) = dQ/dt.

Mathematical translation

An exponential function with argument -t/RC must be differentiated, which is a composition: the outer function is e^u with u = -t/RC.

Step-by-step calculation

Differentiating the outer function with respect to u: d/du(e^u) = e^u. Differentiating the inner function u = -t/RC with respect to t: du/dt = -1/RC. Applying the chain rule: I(t) = Q0 · e^(-t/RC) · (-1/RC) = -(Q0/RC) e^(-t/RC).

Reading the result

The negative sign indicates the current flows in the discharge direction (the charge decreases over time). The magnitude (Q0/RC) e^(-t/RC) decays exponentially, starting at Q0/RC at t=0 and approaching zero as time passes, reflecting that the discharge is fast at first and slows down as the capacitor empties.

Common error avoided

This avoids differentiating only the exponential part while forgetting to multiply by the derivative of the exponent (-1/RC), which would give a result without the correct units and without the correct time scale factor: the chain rule is essential here, not optional.

Inverse example

Given result

The derivative of a position function turns out to be a cosine function with the same angular frequency as the original sine function.

What physical question it can answer

This allows one to recognize that the original position was a sine function of simple harmonic motion, x(t) = A sin(ωt), and that its derivative, velocity v(t) = Aω cos(ωt), is ninety degrees out of phase with the position: when the position is maximal, the velocity is zero, and when the position passes through the equilibrium point, the velocity is maximal.

Common errors

Error 1: forgetting the chain rule in composite functions

It seems reasonable to differentiate e^(-kt) directly as e^(-kt), without multiplying by the derivative of the exponent. It fails because -kt is not the simple variable, but a function of the variable, and the chain rule requires multiplying by its derivative, -k in this case. Detect it by checking whether the function's argument is exactly the variable or an expression depending on it. Correct it by always identifying the inner function before applying the outer rule. Mini-example: the correct derivative of e^(-kt) is -k e^(-kt), not simply e^(-kt).

Error 2: confusing the derivative of sin with that of cos (or vice versa)

It seems reasonable to memorize both rules without paying attention to the sign. It fails because the derivative of sin x is cos x (no negative sign), while the derivative of cos x is -sin x (with a negative sign), and confusing them reverses the predicted physical behavior. Detect it by checking the sign of the result against the expected behavior at known points (for example, where sin x is maximal, its derivative must be zero). Correct it by explicitly memorizing the negative sign associated with the derivative of cosine. Mini-example: differentiating cos(ωt) gives -ω sin(ωt), not ω sin(ωt).

Error 3: applying the power rule to a variable exponent

It seems reasonable to use d/dx(x^n) = nx^(n-1) even when the exponent also depends on x, as in x^x. It fails because that rule assumes n is constant; if the exponent varies, logarithmic differentiation or another different technique is needed. Detect it by checking whether the exponent contains the differentiation variable. Correct it by applying logarithms to both sides before differentiating, instead of forcing the simple power rule. Mini-example: differentiating x^x directly with the power rule gives an incorrect result that ignores the exponent's dependence on x.

Error 4: using degrees instead of radians in trigonometric functions

It seems reasonable to differentiate sin(x) without checking what units x is in. It fails because the standard trigonometric differentiation rules (derivative of sin is cos, no extra factors) are only valid if x is in radians; in degrees, an additional conversion factor (π/180) appears that is frequently forgotten. Detect it by explicitly checking the units of the argument before differentiating. Correct it by converting to radians before applying the standard rules, or including the conversion factor if working in degrees. Mini-example: differentiating sin(x) with x in degrees without the conversion factor gives a numerically incorrect result.

Error 5: differentiating with respect to the wrong variable in a multivariable function

It seems reasonable to differentiate an expression without clearly specifying with respect to which variable it is being differentiated, when the function depends on more than one (for example, position and time in a wave). It fails because the result changes completely depending on the chosen variable. Detect it by explicitly asking which is the differentiation variable requested in the problem. Correct it by always explicitly stating the differentiation variable before applying any rule. Mini-example: differentiating sin(kx - ωt) with respect to x gives a different result than differentiating it with respect to t.

Quick detection rule

If the result of differentiating a composite function does not include the factor corresponding to the derivative of the inner function, if a plus sign appears where a minus sign should be in a trigonometric derivative, or if it is unclear with respect to which variable the differentiation was done, the error is probably in the setup of the applied rule, not in the final arithmetic.

Physical applications

1. Simple harmonic motion

It appears when differentiating positions described with sine or cosine to obtain velocity and acceleration in oscillators. It is calculated using the trigonometric differentiation rules together with the chain rule if there is an angular frequency in the argument. The result gives functions ninety and one hundred eighty degrees out of phase with the position. The dangerous error is forgetting to multiply by the angular frequency when differentiating sin(ωt). As a final control, the learner must explain why acceleration in simple harmonic motion is proportional to position but with opposite sign.

2. Radioactive decay and capacitor discharge

It appears when differentiating exponential functions of the form e^(-kt) to obtain decay or discharge rates. It is calculated using the exponential rule combined with the chain rule. The result gives the instantaneous rate of change, always proportional to the remaining quantity at that instant. The dangerous error is forgetting the negative sign of the exponent when differentiating. As a final control, the learner must explain why the decay rate is highest at first and decreases over time.

3. Sound intensity and logarithmic scales

It appears when differentiating logarithmic functions to relate changes in a linear quantity to changes in its representation on a decibel or other logarithmic scale. It is calculated using the rule d/dx(ln x) = 1/x together with the chain rule. The result shows that a small change on a logarithmic scale corresponds to a large multiplicative change in the original quantity. The dangerous error is treating the logarithmic scale as if it were linear when differentiating. As a final control, the learner must explain why sensitivity to change decreases as the original quantity increases on a logarithmic scale.

4. Electromagnetic and optical waves

It appears when differentiating sinusoidal functions describing oscillating electric and magnetic fields in a wave. It is calculated using the trigonometric rules and the chain rule if there is combined spatial and temporal dependence. The result gives the rate of change of the field at any point and instant. The dangerous error is differentiating with respect to the wrong variable (space instead of time, or vice versa) without identifying which is asked for. As a final control, the learner must explain the physical difference between differentiating a field with respect to time and differentiating it with respect to position.

5. Growth and decay in biological and economic systems

It appears when differentiating exponential growth models of populations or investments with continuous compound interest. It is calculated using the exponential rule. The result shows that the instantaneous rate of change is always proportional to the current size of the quantity. The dangerous error is assuming the rate of change is constant instead of proportional to the current value. As a final control, the learner must explain why an exponentially growing population increases faster and faster in absolute terms, even though its percentage growth rate is constant.

Practice

Level 1: recognition

Identify, for three given functions (t³, e^(-2t), sin(5t)), which basic differentiation rule corresponds to each one.

Exercise: explain why sin(5t) requires the chain rule in addition to the basic trigonometric rule.

Level 2: direct application

Calculate the derivative of simple functions of each type (power, exponential, trigonometric) without composition.

Exercise: calculate the derivative of f(t) = 3t⁴ - 2e^t + cos(t).

Level 3: choosing the tool

Decide what combination of rules (product, quotient, chain) you need to differentiate an expression with composite or combined functions.

Exercise: given the function f(t) = t² sin(3t), decide what rules you need to combine before differentiating.

Level 4: physical interpretation

Interpret the result of a derivative in terms of physical behavior: decay, oscillation, or growth.

Exercise: given the derivative of a capacitor discharge function, explain why its absolute value decreases over time.

Level 5: intentional error

Analyze a calculation where the chain rule was forgotten in a composite function, or where the sign of the sine or cosine derivative was confused. Detect the error, correct it, and explain the physical consequence.

Exercise: invent a plausible error when differentiating an exponential function with a composite exponent, show why it seems reasonable, correct it, and state the incorrect result it would have produced.