Derivative rules cheat sheet: the full first-course list, in words, with worked examples

Finding a derivative from the limit definition is slow, and for some functions it is a rather challenging process⁶. That is why the rules exist. Learn them once and you will seldom go back to the definition¹. Each rule below gets its formula, a plain-words version and a worked example. The Lamar University notes² carry the full proofs.

A cheat sheet for derivatives: the rules at a glance

The logarithm and inverse trig rules follow in a block of their own, because their formulas are fractions. The natural logarithm comes from the exponential⁵, the general logarithm from the natural one¹¹, and the six inverse trig functions from the inverse function theorem¹²:

(ln x)′        = 1/x
(logₐ x)′      = 1 / (x · ln a)
(arcsin x)′    = 1/√(1 − x²)            (arccos x)′ = −1/√(1 − x²)
(arctan x)′    = 1/(1 + x²)             (arccot x)′ = −1/(1 + x²)
(arcsec x)′    = 1/(|x|·√(x² − 1))      (arccsc x)′ = −1/(|x|·√(x² − 1))

Hyperbolic functions (sinh, cosh and relatives) sit at the end of the page under beyond a first course. Three notes apply to every row above and below. Trig formulas assume radians⁴. The power rule needs a variable base and a number exponent¹. And the derivative of a product or quotient is never the product or quotient of the derivatives¹.

The seven core rules, one at a time

Constant rule

The derivative of a constant is zero¹. A constant function is a horizontal line, and the slope, or rate of change, of a horizontal line is zero⁶. A tangent line to the graph of a constant is level, so the slope is flat everywhere⁹.

Constant multiple rule

You can factor a multiplicative constant out of a derivative if you need to¹. In practice you leave the constant where it is and differentiate the rest. The derivative of 5x is 5, because the 5 stays and x becomes 1¹.

Sum and difference rule

To differentiate a sum or a difference, differentiate each term and put the pieces back together with the same signs¹. The rule is not limited to two functions¹. Combined with the power and constant multiple rules, it gives the derivative of any polynomial⁹.

Power rule

Bring the original exponent down in front, multiply, and subtract one from the original exponent¹. The rule requires a variable raised to a number, and the term must be in the numerator¹. Rewrite roots as fractional exponents and move denominators up before you start: when you see radicals, convert them to fractional exponents and simplify the exponents as much as possible¹.

The power rule does not apply to a constant raised to a variable power⁶. For example, 3ˣ needs the exponential rules below, not the power rule⁵.

Product rule

Take the derivative of the first function times the second, then add the first function times the derivative of the second². The derivative of a product is not the product of the derivatives². Early textbook examples can often be done by multiplying out first, so there are few reasons to use the product rule at that stage, but as the functions grow more complicated the product rule becomes more useful and in many cases required².

Quotient rule

The numerator looks like the product rule with a minus sign, over the denominator squared. Be careful not to mix the two numerators up². When the denominator is a plain constant or a single power, rewrite the quotient as a power and skip the quotient rule².

Chain rule

Identify the “inside function” and the “outside function”³. Differentiate the outside function leaving the inside function alone, then multiply by the derivative of the inside function³. The outside function is the last operation you would perform in an evaluation³. When applying the chain rule to a composition, work from the outside function in⁷.

Worked examples

The functions come from the open course notes; the arithmetic is worked here.

Sums, differences and the power rule. A polynomial is handled term by term with the sum rule and the power rule¹:

f(x) = 15x^100 − 3x^12 + 5x − 46
f′(x) = 15·100·x^99 − 3·12·x^11 + 5·1·x^0 − 0
      = 1500x^99 − 36x^11 + 5

Negative exponents. It is an easy mistake to go the other way when subtracting one from a negative exponent¹. The correct step moves the exponent more negative, from −6 to −7:

g(t) = 2t^6 + 7t^(−6)
g′(t) = 12t^5 + 7·(−6)·t^(−7) = 12t^5 − 42t^(−7)

Product rule. For y = x^(2/3)·(2x − x²), the product rule gives a result that can be simplified to match the answer from multiplying first, which is the standard check²:

y′ = (2/3)x^(−1/3)(2x − x²) + x^(2/3)(2 − 2x)
   = (10/3)x^(2/3) − (8/3)x^(5/3)

Quotient rule. For W(z) = (3z + 9)/(2 − z) there is little to do but apply the quotient rule²:

W′(z) = [3(2 − z) − (3z + 9)(−1)] / (2 − z)²
      = 15 / (2 − z)²

Chain rule. For sin(3x² + x) the outside function is the sine and the inside function is 3x² + x³:

d/dx sin(3x² + x) = cos(3x² + x) · (6x + 1)

A chain rule problem often hides a second rule. There can be two terms that each need a different application of the chain rule, so do not expect a single chain rule in a problem³.

Trig derivatives

The derivative of the sine function is the cosine, and the derivative of the cosine function is the negative sine⁸. These formulas depend on two limits, and the proof of the sine formula requires angles in radians⁴. If the angles are in degrees, the limit involving sine is not 1 and the formulas would pick up an extra constant⁴. So always use radians in a calculus class⁴.

The other four need no new memory work. Tangent, cotangent, secant and cosecant are all quotients involving sine, cosine or both, so the quotient rule produces their derivatives⁸. The full set of six is in the table above⁴. Worked from a textbook function:

f(x) = 2 tan x − 3 cot x
f′(x) = 2 sec² x − 3·(−csc² x) = 2 sec² x + 3 csc² x

Why do these matter outside a classroom? Sine and cosine describe simple harmonic motion, which is associated with systems such as an object with mass oscillating on a spring⁸.

Exponential and logarithmic function derivatives

The power rule does not work on an exponential function, because it required the exponent to be a fixed number and the base to be a variable, and the exponential function is exactly the opposite⁵. The natural exponential function is the simple case: eˣ is its own derivative⁵. The natural exponential and natural logarithm are inverses of each other, and that is how the derivative of the logarithm is found from the derivative of the exponential⁵.

For any other base a, with a positive and not equal to one, the same two rules pick up a factor of ln a¹¹:

(aˣ)′ = aˣ · ln a           (logₐ x)′ = 1 / (x · ln a)

Check it with a equal to e: ln e is one, so the extra factor disappears and you are back at the two natural rules. Two quick examples, one with the chain rule:

d/dx 3ˣ         = 3ˣ · ln 3
d/dx log₁₀ x    = 1 / (x · ln 10)
d/dx 2^(5x)     = 2^(5x) · ln 2 · 5

Textbooks state the general case the same way, including the chain-rule version where the exponent or the log argument is itself a function¹¹.

These functions matter because exponential functions play an important role in modeling population growth and the decay of radioactive materials¹¹. When a function is a messy product or quotient with exponents, logarithmic differentiation can determine its derivative: logarithms remove exponents, turn products into sums and division into subtraction¹¹⁹.

Inverse trig derivatives

The derivatives of the inverse trig functions are surprising: they are algebraic functions, with no trig left in them¹². Arcsine, arccosine, arctangent, arccotangent, arcsecant and arccosecant are all in the table above, and the textbook finds the remaining ones from the inverse function theorem¹². Written with a calculator-style name, arcsin x is the same function as sin⁻¹ x, and it is not the reciprocal of sin x. A worked example with the chain rule, from the textbook:

h(x) = arcsin(2x³)
h′(x) = 1/√(1 − (2x³)²) · 6x² = 6x² / √(1 − 4x⁶)

What are the 7 (or 9) rules of derivatives?

The count depends on who is counting. Constant, constant multiple, sum, difference, power, product and quotient make seven; the chain rule makes eight⁶⁷; the trig family and the exponential and logarithm family push a list to nine or more. This page counts seven core rules, with sum and difference as one, and then two families of special functions, so that each group fits in your head.

How to memorize the derivative rules

Memorize the rules in words first and symbols second, because the words carry the logic.

1. Linear rules. Constant, constant multiple, sum and difference. Differentiate each term, keep constants out front¹.
2. Building rules. Power, product, quotient and chain. They handle everything built from simple pieces⁹.
3. Special functions. Sine and cosine, the natural exponential, the natural logarithm. These you simply recall⁸⁵.

Then test yourself on one function per rule until you stop reaching for the page. The reciprocal rule need not be memorized separately, because it can be derived from the quotient rule or from the power and chain rules⁹.

Which rule do I use first?

Look at the outermost operation, the last thing you would do when evaluating the function³.

• Last operation is adding or subtracting: sum and difference rule.
• Last operation is multiplying two functions: product rule.
• Last operation is dividing: quotient rule.
• Last operation is applying a function to an inside expression: chain rule³.

The rules combine, and most real problems use two or three together⁶. In a product of an inverse tangent and a root, for instance, the first step is the product rule, and each derivative then needs a chain rule³. In the reverse case, a chain rule on the outside needs a product rule for the inside³.

Common mistakes

• Wrong direction on a negative exponent. From −6 you subtract one to reach −7, not −5¹.
• Pulling a constant up with the variable. Only the term immediately to the left of the exponent takes it¹.
• Applying the power rule to a constant base. It does not apply to a constant raised to a variable power⁶.
• Differentiating a product piece by piece. The derivative of a product is not the product of the derivatives².
• Mixing up the two numerators. The quotient rule’s numerator looks like the product rule’s, so check the minus sign².
• Differentiating the inside too early. Differentiate the outside function and leave the inside alone until the final multiplication³.
• Degrees instead of radians. The trig derivatives change if angles are in degrees⁴.
• Logs and products. If a function is a messy product or quotient with exponents, logarithmic differentiation can simplify it first: logarithms remove exponents, turn products into sums and division into subtraction⁹.

Beyond a first course: hyperbolic and inverse (arc) hyperbolic derivatives

You can skip this section until a course asks for it. Hyperbolic functions are built from the exponential. The hyperbolic sine and hyperbolic cosine are defined from the exponential¹⁰, and the other four come from those two¹⁰:

sinh x = (eˣ − e⁻ˣ) / 2        cosh x = (eˣ + e⁻ˣ) / 2
tanh x = sinh x / cosh x        coth x = cosh x / sinh x
sech x = 1 / cosh x             csch x = 1 / sinh x

Their derivatives look like the trig table with different signs¹⁰:

The inverse hyperbolic functions follow the same implicit-differentiation route as the inverse trig ones¹⁰, and the textbook tabulates the results, with the range each one needs¹⁰:

(sinh⁻¹ x)′ = 1/√(x² + 1)             all x
(cosh⁻¹ x)′ = 1/√(x² − 1)             x above one
(tanh⁻¹ x)′ = 1/(1 − x²)              x between −1 and 1
(coth⁻¹ x)′ = 1/(1 − x²)              x beyond ±1
(sech⁻¹ x)′ = −1/(x·√(1 − x²))        x between 0 and 1
(csch⁻¹ x)′ = −1/(|x|·√(1 + x²))      x not zero

The tanh and coth rows match, which is why integrating that fraction needs the domain to pick the right antiderivative¹⁰. Combine any of them with the chain rule when the argument is an expression instead of a bare x⁷.

See the rules at work in a normal morning, or in the exponential behind a loan payment. The rest of the core ideas section grows from this sheet.