Differentiation rules

Computing every derivative from the limit definition would be slow and error-prone. Fortunately, a small set of rules lets you differentiate almost any function an engineer will meet: polynomials, products and quotients, compositions, exponentials, logarithms and trigonometric functions. Each rule can be proved from the definition in Lesson 2, and seeing why a rule works makes it much easier to apply correctly.

By the end of this lesson you should be able to differentiate complicated expressions quickly and reliably, choose the right rule for each part of an expression, and handle equations where \(y\) is not given explicitly as a function of \(x\).

The basic rules

Constants and the power rule

A constant never changes, so its derivative is zero: \(\dfrac{d}{dx} c = 0\). For powers of \(x\), the power rule is

It holds for every real exponent \(n\): positive, negative and fractional. You saw three cases in Lesson 2: \(x^2 \to 2x\), \(x^{-1} \to -x^{-2}\) and \(x^{1/2} \to \tfrac12 x^{-1/2}\).

To use the power rule on roots and fractions, first rewrite them as powers. \(\sqrt{x} = x^{1/2}\), \(\sqrt[3]{x^2} = x^{2/3}\), and \(\dfrac{1}{x^3} = x^{-3}\).

Constant multiples, sums and differences

Constants carry through, and derivatives of sums are sums of derivatives:

Together with the power rule, these let you differentiate any polynomial term by term.

The product rule

The derivative of a product is not the product of the derivatives. Instead:

In words: derivative of the first times the second, plus the first times derivative of the second.

Why it works: in the difference quotient for \(fg\), add and subtract \(f(x+h)g(x)\) on top:

As \(h \to 0\), \(f(x+h) \to f(x)\) and the two difference quotients become \(g'(x)\) and \(f'(x)\).

A rectangle gives a physical picture. If a plate has sides \(f\) and \(g\) that are both growing, its area \(A = fg\) grows in two strips: one of width \(f'\) along side \(g\), and one of width \(g'\) along side \(f\). The tiny corner piece shrinks away in the limit.

The quotient rule

For a quotient:

The order in the numerator matters because of the minus sign. A common memory aid: "low d-high minus high d-low, over the square of what's below."

The chain rule

The chain rule differentiates compositions, functions inside functions, and it is the rule you will use most. If \(y = f(u)\) and \(u = g(x)\), then

A practical way to apply it: identify the outside function and the inside function. Differentiate the outside, leaving the inside alone, then multiply by the derivative of the inside.

The chain rule applies repeatedly to longer chains. Work from the outside in, multiplying a derivative at each layer.

Exponential and logarithmic functions

The number \(e \approx 2.71828\) is chosen so that the exponential function is its own derivative:

Combined with the chain rule, these give \(\dfrac{d}{dx} e^{g(x)} = e^{g(x)} g'(x)\) and \(\dfrac{d}{dx}\ln g(x) = \dfrac{g'(x)}{g(x)}\). For other bases, \(\dfrac{d}{dx} a^x = a^x \ln a\) and \(\dfrac{d}{dx}\log_a x = \dfrac{1}{x\ln a}\).

Exponentials describe anything whose rate of change is proportional to its current size: radioactive decay, cooling, charging and discharging capacitors, bacterial growth, and the pressure drop in a draining tank. In each case the derivative brings out the same exponential, multiplied by a rate constant.

Trigonometric functions

With angles in radians, the derivatives of the trigonometric functions are:

The first two follow from the definition and the limits \(\dfrac{\sin x}{x} \to 1\) and \(\dfrac{1 - \cos x}{x} \to 0\) from Lesson 1. The third follows from the quotient rule applied to \(\sin x / \cos x\). Others you will occasionally need: \(\dfrac{d}{dx}\sec x = \sec x\tan x\), \(\dfrac{d}{dx}\csc x = -\csc x\cot x\), and \(\dfrac{d}{dx}\cot x = -\csc^2 x\).

These formulas only hold in radians. In degrees an extra factor of \(\pi/180\) appears in every derivative, which is why engineering calculus always uses radians.

Vibrations and oscillations

An object in simple harmonic motion (a mass on a spring, a vibrating machine part, alternating current) has position \(x(t) = A\cos(\omega t)\), where \(A\) is the amplitude and \(\omega\) the angular frequency in rad/s. Differentiating twice with the chain rule:

The result \(a = -\omega^2 x\) is the defining equation of simple harmonic motion: acceleration proportional to displacement and directed back toward equilibrium.

Notice how frequency amplifies the derivatives: velocity scales with \(\omega\) and acceleration with \(\omega^2\). A tiny displacement at high frequency can mean a large acceleration, which is why high-frequency vibration damages bearings and electronics.

Implicit differentiation

Sometimes \(y\) is defined by an equation instead of an explicit formula, such as \(x^2 + y^2 = 25\) for a circle. You can still find \(dy/dx\): differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\), so every derivative of a \(y\) term picks up a factor \(dy/dx\) by the chain rule. Then solve for \(dy/dx\).

Logarithmic differentiation

For functions with variable exponents, or long products and quotients, take the natural logarithm of both sides first, then differentiate implicitly.

Inverse trigonometric functions

Implicit differentiation also gives the derivatives of the inverse trig functions, which appear when integrating and in geometry problems:

For example, if \(y = \arctan x\), then \(\tan y = x\). Differentiating gives \(\sec^2 y \cdot y' = 1\), so \(y' = \cos^2 y = \dfrac{1}{1 + \tan^2 y} = \dfrac{1}{1 + x^2}\).

Hyperbolic functions and hanging cables

One more family of functions appears often enough in engineering to learn now. The hyperbolic sine and cosine are built from exponentials:

They share many identities with the trigonometric functions, with a few sign changes; the most useful is \(\cosh^2 x - \sinh^2 x = 1\). Their derivatives follow directly from the exponential rule:

Note there is no minus sign, unlike \(\cos x\). Check it yourself: differentiating \(\tfrac12(e^x + e^{-x})\) gives \(\tfrac12(e^x - e^{-x})\).

A cable or chain hanging freely under its own weight takes the shape of a catenary, \(y = a\cosh(x/a)\), where \(a\) is the ratio of the horizontal tension to the weight per unit length. Power lines, suspension-bridge cables before the deck is hung, and hanging chains all follow this curve. The derivative gives the cable's slope, which determines the direction of the tension at the supports.

Hyperbolic functions also appear in heat transfer through fins, in the deflection of beams on elastic foundations, and in the analysis of long transmission lines.

Choosing the right rule

Most real expressions need several rules at once. Decide what the expression is at its outermost level, apply that rule, and repeat on the pieces.

• A sum or difference: differentiate term by term.
• A constant times something: keep the constant and differentiate the rest.
• A product of two non-constant factors: product rule.
• A quotient: simplify if you can; otherwise use the quotient rule.
• A function of something other than plain \(x\): chain rule.
• A variable in an exponent with a variable base, or a tangled product: logarithmic differentiation.