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.

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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 $C$ , while derivatives often require attention to chain rule factors such as $u'$ .

Key Facts

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

Vocabulary

Derivative: A derivative gives the instantaneous rate of change of a function, written as $f'(x)$ or $\frac{dy}{dx}$ .
Antiderivative: An antiderivative of $f(x)$ is a function $F(x)$ such that $F'(x)=f(x)$ .
Indefinite Integral: An indefinite integral, written $\int f(x)\,dx$ , represents the family of all antiderivatives of $f(x)$ .
Constant of Integration: The constant of integration $C$ represents all vertical shifts of an antiderivative because the derivative of any constant is $0$ .
Chain Rule: The chain rule differentiates composite functions using $\frac{d}{dx}\left(f(g(x))\right)=f'(g(x))g'(x)$ .
Substitution: Substitution rewrites an integral using $u=g(x)$ and $du=g'(x)\,dx$ to match a simpler antiderivative rule.

Common Mistakes to Avoid

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

Practice Questions

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

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