Differentiation Rules
The derivative measures how fast a function changes: the slope of its graph at each point. Seven rules cover most functions you will meet.
- Power rule
- Product rule
- Quotient rule
- Chain rule
- Trigonometric functions
- Exponential functions
- Logarithmic functions
Which rule?
Look at the outermost structure of the function.
- A sum or difference: differentiate term by term.
- A product of two functions of x: product rule. A ratio: quotient rule.
- A function inside a function: chain rule.
- Otherwise match a basic form: power, trigonometric, exponential or logarithmic.
Power rule
Use it when the variable is raised to a constant power. Roots and reciprocals count too: rewrite \sqrt{x} as x^{1/2} and \tfrac{1}{x^3} as x^{-3}. Bring the exponent down, then lower it by one.
Example. Differentiate x^3+2x.
Do each term: \tfrac{d}{dx}x^3=3x^2 and \tfrac{d}{dx}2x=2. The derivative is 3x^2+2.
Watch out: the power rule is for x^n (variable in the base). For 2^x the variable is in the exponent: see exponential functions.
Product rule
Use it when two functions of x are multiplied and neither is just a constant. Take the derivative of one factor at a time and keep the other.
Example. Differentiate x\sin x.
Let u=x and v=\sin x, so u'=1 and v'=\cos x. Then u'v+uv'=\sin x+x\cos x.
Watch out: the derivative of a product is not the product of the derivatives.
Quotient rule
Use it when one function is divided by another. Remember the order: bottom times derivative of top, minus top times derivative of bottom, all over bottom squared.
Example. Differentiate \dfrac{x^2+1}{x-1}.
With u=x^2+1 and v=x-1: u'=2x, v'=1. So \dfrac{2x(x-1)-(x^2+1)\cdot1}{(x-1)^2}=\dfrac{x^2-2x-1}{(x-1)^2}.
Watch out: the subtraction is not symmetric. Swapping the two products flips the sign of the answer.
Chain rule
Use it when one function sits inside another, such as \sin(x^2), (3x+1)^5 or e^{2x}. Differentiate the outer function, leave the inside alone, then multiply by the derivative of the inside.
Example. Differentiate \sin(x^2).
- Outer: \sin u, inner: u=x^2.
- Outer derivative: \cos u. Inner derivative: 2x.
- Multiply: 2x\cos(x^2).
Watch out: forgetting the inner derivative is the most common slip.
Trigonometric functions
Use it when sine, cosine or tangent appears. Learn three results; the rest follow from the product, quotient and chain rules. Angles are in radians.
Example. Differentiate \sin x+\cos x.
Term by term: \cos x+(-\sin x). The derivative is \cos x-\sin x.
Watch out: cosine differentiates to minus sine. If the angle is not plain x, as in \sin 3x, add the chain rule: 3\cos 3x.
Exponential functions
Use it when the variable is in the exponent. The function e^x is special because it is its own derivative. For another base a, a factor \ln a appears.
Example. Differentiate e^{3x}.
The inside 3x has derivative 3, so the chain rule gives 3e^{3x}.
Watch out: do not use the power rule here. x^2 and 2^x are different kinds of function.
Logarithmic functions
Use it when the function contains a natural logarithm. In the calculator, ln and log both mean the natural logarithm. With something inside, the chain rule puts that inside on top: the derivative of the inside over the inside.
Example. Differentiate \ln(x^2+1).
The inside is x^2+1 with derivative 2x, so the answer is \dfrac{2x}{x^2+1}.
Watch out: simplify first when you can: \ln(x^3)=3\ln x, which differentiates to \tfrac{3}{x}.
Where derivatives are used
Derivatives give velocity from position, find the highest and lowest points of a curve, and measure how sensitive one quantity is to another. The last step of any derivative is to check that you used the rules in the right order.