Calculus Formula Sheet

Differentiation answers "how fast is this changing right now," and integration answers "how much has accumulated." The fastest way to keep calculus formulas straight is to learn them as paired sets, with their symbols and exceptions, rather than as a flat list to memorize.

The Core Formulas And Their Symbols

In every formula below, $a$ , $b$ , and $c$ are constants, $f$ and $g$ are differentiable functions, and $C$ is the constant of integration.

Basic Differentiation

\frac{d}{dx} c = 0

\frac{d}{dx} x^n = n x^{n-1}

\frac{d}{dx} \left(af(x) + bg(x)\right) = af'(x) + bg'(x)

Polynomials can be differentiated term by term. For products, quotients, and composite functions:

\frac{d}{dx} \left(f(x)g(x)\right) = f'(x)g(x) + f(x)g'(x)

\frac{d}{dx} \left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}, \quad g(x) \ne 0

\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)

For nested forms like $(2x+1)^5$ or $\sin(3x)$ , the chain rule is indispensable.

Basic Integration

\int c \, dx = cx + C

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \ne -1

\int \left(af(x) + bg(x)\right)\,dx = a\int f(x)\,dx + b\int g(x)\,dx

\int \frac{1}{x} \, dx = \ln|x| + C

The trigonometric, exponential, and logarithmic forms used most often are:

\frac{d}{dx} \sin x = \cos x, \qquad \frac{d}{dx} \cos x = -\sin x

\frac{d}{dx} e^x = e^x, \qquad \frac{d}{dx} \ln x = \frac{1}{x}, \quad x > 0

\int e^x \, dx = e^x + C, \qquad \int \sin x \, dx = -\cos x + C, \qquad \int \cos x \, dx = \sin x + C

Why These Formulas Hold

The power rule comes from the fact that increasing the exponent multiplies in one extra factor of the variable, so its rate of change carries the exponent down as a coefficient and drops the power by one. Integration runs that machine in reverse: it raises the exponent by one and divides by the new exponent, which is exactly what undoes the power rule.

That reverse view explains the famous exception. When $n = -1$ , raising the exponent gives $x^0$ and dividing by $n+1 = 0$ is undefined, so $\int x^{-1}\,dx$ cannot follow the pattern. Instead it equals $\ln|x| + C$ , precisely because $\frac{d}{dx}\ln|x| = \frac{1}{x}$ . The $+C$ appears for the same structural reason: since the derivative of any constant is $0$ , an antiderivative is only pinned down up to an added constant.

Worked Example: Differentiate And Integrate Term By Term

Consider:

f(x) = 2x^3 - 3x^2 + 4x - 1

Differentiating term by term, bring each exponent down and reduce it by one:

f'(x) = 6x^2 - 6x + 4

Now take the indefinite integral of the same expression, raising each exponent by one:

\int \left(2x^3 - 3x^2 + 4x - 1\right)\,dx = \frac{1}{2}x^4 - x^3 + 2x^2 - x + C

Differentiation lowers the exponent by one; integration raises it by one. But because integration adds $+C$ , it is an inverse "with a constant margin," not a perfect one-to-one undo.

Practice It Yourself

Differentiate $f(x) = 3x^4 - 2x + 7$ , then find the indefinite integral of that same expression. Once the polynomial cases feel automatic, differentiate $(3x+1)^4$ to rehearse a chain-rule case, and check your answer by differentiating the result back to the original. If you want more reps, pick an expression with a trigonometric or composite function and decide which formulas it needs before computing.

Calculation Pitfalls To Watch

Plugging $n = -1$ directly into the integral power rule. Remember that $\int \frac{1}{x}\,dx = \ln|x| + C$ .
Forgetting to multiply by the inner derivative when differentiating composites like $(2x+1)^5$ . This is the classic chain-rule slip.
Forgetting $+C$ on an indefinite integral. It is required every time.
Swapping the signs of $\int \sin x \, dx$ and $\int \cos x \, dx$ . When unsure, differentiate the result to check.
Differentiating a product or quotient term by term instead of using the product or quotient rule.

Differentiation formulas give slopes, velocity, acceleration, and extreme values; integration formulas give area, distance traveled, and accumulation. Seeing each formula as a tool for moving between "how it is changing now" and "how much has accumulated" makes choosing the right one far more intuitive.

Frequently Asked Questions

What are the basic differentiation formulas to memorize first?: Start with the derivative of a constant being 0, the power rule turning x to the n into n times x to the n minus 1, and linearity for sums with constant coefficients. Then add the product rule, the quotient rule for nonzero denominators, and the chain rule for composite functions.
What is the exception to the integration power rule?: The power rule for integrals, x to the n integrating to x to the n plus 1 over n plus 1 plus C, fails when n equals negative 1, because that would divide by zero. In that case the integral of 1 over x is the natural log of the absolute value of x, plus C.
When do you need the chain rule?: Use the chain rule whenever functions are nested inside each other, such as the quantity 2x plus 1 raised to the fifth power or sine of 3x. The rule says the derivative of f of g of x equals f prime of g of x times g prime of x, and it is indispensable for these composite forms.
Why do you add plus C to an indefinite integral?: Because the derivative of any constant is 0, an antiderivative is only determined up to an added constant, so indefinite integrals carry a plus C. It is easy to forget, so make writing the constant a habit every time you compute an indefinite integral.
How can you check an integration answer?: Differentiate your result and see whether you get back the original function. This catches the most frequent slips, especially sign errors with the sine and cosine integrals, where the integral of sine x is negative cosine x plus C and the integral of cosine x is sine x plus C.

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