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Common Derivatives & Integrals Table cheat sheet - grade 11-12

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Calculus Grade 11-12

Common Derivatives & Integrals Table Cheat Sheet

A printable reference covering derivative rules, antiderivative rules, power, exponential, logarithmic, trigonometric, inverse trigonometric, substitution, and integration constants for grades 11-12.

TopicsDerivativesIntegralsFunctionsTrigonometryExponentsCalculusStudy Guide
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This cheat sheet covers the most common derivative and integral formulas used in high school calculus. Students need these formulas to recognize patterns quickly, check work, and solve problems involving rates of change and accumulation. It is especially useful for homework, test review, and building fluency before studying applications of calculus.

The core idea is that differentiation and integration are inverse processes, but each has rules that must be applied carefully. Important formulas include the power rule, exponential and logarithmic rules, trigonometric rules, and inverse trigonometric integrals. Antiderivatives usually require a constant CC, while derivatives often require attention to chain rule factors such as u′u'.

Key Facts

  • The power rule for derivatives is ddx(xn)=nxn−1\frac{d}{dx}\left(x^n\right)=nx^{n-1} for any real number nn where the expression is defined.
  • The power rule for antiderivatives is ∫xn dx=xn+1n+1+C\int x^n\,dx=\frac{x^{n+1}}{n+1}+C for n≠−1n\neq -1.
  • The derivative of the natural logarithm is ddx(ln⁡x)=1x\frac{d}{dx}\left(\ln x\right)=\frac{1}{x} for x>0x>0, and ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx=\ln|x|+C.
  • The exponential rules are ddx(ex)=ex\frac{d}{dx}\left(e^x\right)=e^x and ∫ex dx=ex+C\int e^x\,dx=e^x+C.
  • The basic sine and cosine rules are ddx(sin⁡x)=cos⁡x\frac{d}{dx}\left(\sin x\right)=\cos x, ddx(cos⁡x)=−sin⁡x\frac{d}{dx}\left(\cos x\right)=-\sin x, ∫cos⁡x dx=sin⁡x+C\int \cos x\,dx=\sin x+C, and ∫sin⁡x dx=−cos⁡x+C\int \sin x\,dx=-\cos x+C.
  • The tangent and secant squared pair is ddx(tan⁡x)=sec⁡2x\frac{d}{dx}\left(\tan x\right)=\sec^2 x and ∫sec⁡2x dx=tan⁡x+C\int \sec^2 x\,dx=\tan x+C.
  • For a composite function, the chain rule is ddx(f(g(x)))=f′(g(x))g′(x)\frac{d}{dx}\left(f(g(x))\right)=f'(g(x))g'(x), and the matching integral idea is substitution with u=g(x)u=g(x) and du=g′(x) dxdu=g'(x)\,dx.
  • Common inverse trigonometric forms include ∫11−x2 dx=arcsin⁡x+C\int \frac{1}{\sqrt{1-x^2}}\,dx=\arcsin x+C and ∫11+x2 dx=arctan⁡x+C\int \frac{1}{1+x^2}\,dx=\arctan x+C.

Vocabulary

Derivative
A derivative gives the instantaneous rate of change of a function, written as f′(x)f'(x) or dydx\frac{dy}{dx}.
Antiderivative
An antiderivative of f(x)f(x) is a function F(x)F(x) such that F′(x)=f(x)F'(x)=f(x).
Indefinite Integral
An indefinite integral, written ∫f(x) dx\int f(x)\,dx, represents the family of all antiderivatives of f(x)f(x).
Constant of Integration
The constant of integration CC represents all vertical shifts of an antiderivative because the derivative of any constant is 00.
Chain Rule
The chain rule differentiates composite functions using ddx(f(g(x)))=f′(g(x))g′(x)\frac{d}{dx}\left(f(g(x))\right)=f'(g(x))g'(x).
Substitution
Substitution rewrites an integral using u=g(x)u=g(x) and du=g′(x) dxdu=g'(x)\,dx to match a simpler antiderivative rule.

Common Mistakes to Avoid

  • Forgetting the constant CC in an indefinite integral is wrong because ∫f(x) dx\int f(x)\,dx represents a whole family of functions, not just one function.
  • Using the power rule on ∫x−1 dx\int x^{-1}\,dx is wrong because the formula ∫xn dx=xn+1n+1+C\int x^n\,dx=\frac{x^{n+1}}{n+1}+C does not apply when n=−1n=-1.
  • Dropping the chain rule factor is wrong because ddx(sin⁡(3x))\frac{d}{dx}\left(\sin(3x)\right) equals 3cos⁡(3x)3\cos(3x), not just cos⁡(3x)\cos(3x).
  • Confusing derivative and integral signs for trigonometric functions is wrong because ddx(cos⁡x)=−sin⁡x\frac{d}{dx}\left(\cos x\right)=-\sin x while ∫cos⁡x dx=sin⁡x+C\int \cos x\,dx=\sin x+C.
  • Ignoring absolute value in logarithmic antiderivatives is wrong because ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx=\ln|x|+C, which works for both positive and negative values of xx.

Practice Questions

  1. 1 Find ddx(5x4−3x2+7x−9)\frac{d}{dx}\left(5x^4-3x^2+7x-9\right).
  2. 2 Evaluate ∫(6x2−4x+1) dx\int \left(6x^2-4x+1\right)\,dx.
  3. 3 Find ∫2x1+x2 dx\int \frac{2x}{1+x^2}\,dx using substitution.
  4. 4 Explain why every indefinite integral answer needs +C+C, even when the antiderivative formula looks complete.

Understanding Common Derivatives & Integrals Table

A formula table is most useful after you identify the structure of an expression. Start by finding the outside operation. In the expression sine of three x squared, sine is outside and three x squared is inside.

The derivative begins with cosine of three x squared, then it must include the rate of change of the inside expression. This is the chain rule in action. Missing that extra factor is one of the most common errors in calculus.

For integrals, work in the reverse direction. Look for a function and a nearby factor that could be the derivative of its inside expression.

The power rule has an important exception. An expression with x to the negative one power does not follow the usual antiderivative power pattern because the new exponent would create division by zero. That special form leads to a logarithm instead.

This is a reminder that tables are not just lists to memorize. Each rule has conditions. Logarithms require careful attention to where an expression is positive or negative.

Square roots restrict possible input values. Inverse trigonometric forms often appear only after an algebra step makes the denominator match the required pattern.

Differentiation and integration describe different physical ideas. A derivative can represent instantaneous velocity from position, the changing temperature of a cooling drink, or the growth rate of a population. An integral can represent total distance from velocity, total water entering a tank from a flow rate, or total electric charge from current.

Units provide a strong check. If velocity is measured in meters per second, integrating over seconds should give meters. If a result has units that do not fit the situation, the setup or calculation needs review.

A reliable way to check an antiderivative is to differentiate your answer. The result should return the original integrand. This check catches sign errors, missing constants, and incorrect chain rule factors.

The constant of integration matters because many different functions have the same derivative. A definite integral does not keep this arbitrary constant because subtracting the endpoint values removes it. When studying a table, group formulas by patterns rather than memorizing isolated lines.

Notice which functions stay nearly unchanged, which switch signs, and which require an inner derivative. With practice, the table becomes a guide for reasoning instead of a page to copy from.