Most derivatives you meet in engineering are not derivatives of a single basic function. They are sums, products, quotients and nested compositions of basic functions such as xn, ex, sin x and ln x. So you need two things to differentiate them. The first is the derivative of each basic function. The second is a small set of rules for combining those derivatives, and this page collects that second set. Each rule is shown as a function h(x) on the left and its derivative h'(x) on the right. With these five rules, any expression built from the basic functions can be differentiated step by step.
Rule of Linear Combination
The first question is what happens to a derivative when functions are scaled and added. The answer is the simplest of the five rules, and every other rule on this page relies on it.
The equation below starts from h(x) = a f(x) + b g(x), where a and b are constants. The derivative is taken term by term, and each constant stays in front of its own term.
The derivative of a weighted sum is the same weighted sum of the derivatives.
The second term in the equation carries the wrong constant : the picture writes a in front of d/dx g(x) and a in front of g'(x). Both should be b. So the correct result is h'(x) = a f'(x) + b g'(x).Differentiation is a linear operation : scaling a function scales its derivative, and adding functions adds their derivatives. This is why a polynomial can be differentiated one term at a time.Subtraction is included : a difference f(x) - g(x) is the case a = 1 and b = -1, so its derivative is f'(x) - g'(x).A worked example : for h(x) = 3x2 + 5 sin x, the rule gives h'(x) = 6x + 5 cos x.A constant term disappears : a constant c has zero slope everywhere. So h(x) = f(x) + c has the same derivative as f(x).
Product Rule
When two functions are multiplied, the derivative is not the product of the two derivatives. This is the most common mistake with products, so let's see why the correct rule has two terms.
The equation below starts from h(x) = f(x) g(x). The derivative lets f(x) change while g(x) is held as it is, then lets g(x) change while f(x) is held, and adds the two results.
Each factor takes its turn to change, while the other factor is held as it is.
Two terms, not one : h'(x) = f'(x) g(x) + f(x) g'(x). Writing f'(x) g'(x) instead fails even for h(x) = x times x. There the correct derivative is 2x, and the wrong formula gives 1.Why two terms appear : think of f g as the area of a rectangle with sides f and g. When x moves by a small step dx, side f grows by f' dx and side g grows by g' dx. The area gains two thin strips, g f' dx and f g' dx. The small corner piece f' g' dx2 vanishes as dx goes to zero.A worked example : for h(x) = x2 ex, h'(x) = 2x ex + x2 ex = x(x + 2) ex. At x = 1 this is 3e, about 8.15.More than two factors : apply the rule twice. Each term differentiates one factor and keeps the others, so the derivative of f g k is f' g k + f g' k + f g k'.
Chain Rule
Many engineering functions are nested, such as sin(ωt) or e-at. The chain rule answers how fast the outer function changes when its input is itself changing with x.
The equation below starts from h(x) = f(g(x)). The derivative is the rate of change of f with respect to its input g, multiplied by the rate of change of g with respect to x. The last form writes the same product with dh/dg in place of df/dg.
Rates of change multiply along a chain of nested functions.
Differentiate the outer function at the inner value : f'(g(x)) is the derivative of f evaluated at g(x), not at x. Then multiply it by g'(x).A worked example : for h(x) = sin(x2), the outer function is sin and the inner function is x2. So h'(x) = cos(x2) times 2x.A second example : for h(x) = (3x + 1)5, h'(x) = 5(3x + 1)4 times 3, which is 15(3x + 1)4.The Leibniz form is a memory aid : in dh/dg times dg/dx, the dg looks as if it cancels. It is not a real cancellation. But the pattern extends to longer chains, dh/dx = dh/du times du/dv times dv/dx.A familiar case : for sin(ωt), the inner derivative is ω, so the derivative is ω cos(ωt). This is why differentiating a sinusoid scales its amplitude by its angular frequency.
Reciprocal Rule
The reciprocal rule covers h(x) = 1/f(x). It is not a new idea. It is the chain rule applied with the outer function 1/u, whose derivative is -1/u2.
The equation below differentiates 1/f(x) in two forms. The middle form shows the chain rule at work: -1/[f(x)]2 is the outer derivative, and df/dx is the inner one. The right side collects them into one fraction.
The reciprocal changes sign and divides by the square of the original function.
The result always carries a minus sign : where f(x) grows, 1/f(x) shrinks. So the two derivatives always have opposite signs.A worked example : for h(x) = 1/(x2 + 1), f'(x) = 2x, so h'(x) = -2x/(x2 + 1)2. At x = 1 this is -1/2.A check against the power rule : for f(x) = x, the rule gives the derivative of 1/x as -1/x2. This matches the power rule applied to x-1.Watch the zeros of f : the rule holds only where f(x) is not zero, because neither h(x) nor h'(x) exists there.Relative changes simply flip sign : dividing h'(x) by h(x) gives -f'(x)/f(x). So when f rises by 1 percent, 1/f falls by about 1 percent. For example, a resistance that rises by 1 percent lowers the conductance by about 1 percent.
Quotient Rule
The quotient rule covers h(x) = f(x)/g(x), and it is the rule where the order of the terms matters most. It is the product rule applied to f(x) and 1/g(x), with the reciprocal rule for the second factor.
The equation below writes the derivative with d/dx notation in the middle and with primes on the right. The numerator is the derivative of the top times the bottom, minus the derivative of the bottom times the top. The denominator is the bottom function squared.
The numerator differs from the product rule only by its minus sign, so the order of its two terms decides the sign of the result.
It follows from the two rules above : write f/g as f times 1/g. The product rule and the reciprocal rule give f'/g - f g'/g2. Over a common denominator, this is (f' g - g' f)/g2.The order of the numerator matters : swapping the two terms flips the sign of the whole result. Start from the derivative of the top to avoid that mistake.A worked example : for h(x) = tan x = sin x / cos x, h'(x) = (cos x cos x + sin x sin x)/cos2 x, which is 1/cos2 x.A second example : for h(x) = sin x / x, h'(x) = (x cos x - sin x)/x2.The reciprocal rule is a special case : with f(x) = 1, f'(x) = 0 and the quotient rule reduces to -g'(x)/[g(x)]2.