Difficulty: Introductory to Intermediate | Prerequisites: Precalculus (algebra, trigonometry, exponentials, logarithms)
Calculus 1 is built on two operations: differentiation (finding instantaneous rates of change) and integration (finding accumulated quantities). Every formula on this page is a tool for carrying out one of those two operations on a specific family of functions, whether polynomial, trigonometric, exponential, or logarithmic. If you are comfortable with algebra and basic trig identities, you have what you need to follow along. The formulas here form the mechanical backbone of the course; nearly every problem you encounter will come down to recognising which rule applies and executing it cleanly.
This covers every derivative and integral formula you need for Calc 1: the power rule, product/quotient/chain rules, all six trig functions and their inverses, exponentials, logarithms, and key trig identities. It also includes integration by parts. If you can recognise which formula to apply, you can solve most problems on the exam.
Derivative
The instantaneous rate of change of a function with respect to its variable, written d/dx f(x) or f'(x). Think of it as the slope of the curve at a single point.
Integral (indefinite)
The family of antiderivatives of a function, written with the integral sign and always including + C. In simple terms, this means reversing a derivative to recover the original function, up to a constant.
Power rule
The formula d/dx (x^n) = n x^(n-1). In simple terms, bring the exponent down as a coefficient and reduce the exponent by one.
Product rule
The formula for differentiating a product of two functions: d/dx [f(x) g(x)] = f(x) g'(x) + f'(x) g(x). Think of it as: first times derivative of second, plus derivative of first times second.
Quotient rule
The formula for differentiating a fraction of two functions. The result is (denominator times derivative of numerator, minus numerator times derivative of denominator) all over denominator squared. Think of it as "low d-high minus high d-low, over low squared."
Chain rule
The formula for differentiating a composition of functions: d/dx f(g(x)) = f'(g(x)) g'(x). In simple terms, differentiate the outer function, leave the inner function alone, then multiply by the derivative of the inner function.
Integration by parts
A technique for integrating a product of two functions, based on the formula: integral of u dv = uv minus integral of v du. Think of it as the reverse of the product rule for derivatives.
Power rule
d/dx (x^n) = n x^(n-1)
Corresponding integral: integral of x^n dx = x^(n+1) / (n+1) + C, provided n is not equal to -1
Example: d/dx (x^4) = 4x^3
Product rule
d/dx [f(x) g(x)] = f(x) g'(x) + f'(x) g(x)
Example: d/dx [x^2 sin x] = x^2 cos x + 2x sin x
Quotient rule
d/dx [f(x) / g(x)] = [g(x) f'(x) - f(x) g'(x)] / [g(x)]^2
Mnemonic: "low d-high minus high d-low, over low squared"
Example: d/dx [sin x / x^2] = [x^2 cos x - 2x sin x] / x^4
Reciprocal rule
d/dx [1 / g(x)] = -g'(x) / [g(x)]^2
This is the quotient rule with f(x) = 1. Useful as a shortcut when only the denominator varies.
Chain rule
d/dx f(g(x)) = f'(g(x)) g'(x)
In words: differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function.
Example: d/dx (sin(x^2)) = cos(x^2) 2x
All six trig functions and their derivative/integral pairs. These come up constantly, so know them cold.
sin x
Derivative: d/dx (sin x) = cos x
Integral: integral of sin x dx = -cos x + C
cos x
Derivative: d/dx (cos x) = -sin x
Integral: integral of cos x dx = sin x + C
tan x
Derivative: d/dx (tan x) = sec^2 x
Integral: integral of tan x dx = ln|sec x| + C
cot x
Derivative: d/dx (cot x) = -csc^2 x
Integral: integral of cot x dx = ln|sin x| + C
sec x
Derivative: d/dx (sec x) = sec x tan x
Integral: integral of sec x dx = ln|sec x + tan x| + C
csc x
Derivative: d/dx (csc x) = -csc x cot x
Integral: integral of csc x dx = ln|csc x - cot x| + C
Note the pattern: the three "co" functions (cos, cot, csc) all pick up a negative sign in their derivatives.
These show up in integration problems more than in straightforward differentiation. Learn the derivative forms so you can recognise the integral patterns.
arcsin (sin^-1)
Derivative: d/dx (sin^-1 x) = 1 / sqrt(1 - x^2)
Related integral: integral of 1 / sqrt(a^2 - u^2) du = sin^-1(u/a) + C
arccos (cos^-1)
Derivative: d/dx (cos^-1 x) = -1 / sqrt(1 - x^2)
Note: this is just the negative of the arcsin derivative. Rarely tested separately.
arctan (tan^-1)
Derivative: d/dx (tan^-1 x) = 1 / (1 + x^2)
Related integral: integral of 1 / (a^2 + u^2) du = (1/a) tan^-1(u/a) + C
arccot (cot^-1)
Derivative: d/dx (cot^-1 x) = -1 / (1 + x^2)
arcsec (sec^-1)
Derivative: d/dx (sec^-1 x) = 1 / (|x| sqrt(x^2 - 1))
Related integral: integral of 1 / (u sqrt(u^2 - a^2)) du = (1/a) sec^-1(u/a) + C
arccsc (csc^-1)
Derivative: d/dx (csc^-1 x) = -1 / (|x| sqrt(x^2 - 1))
These identities are used constantly to simplify expressions before differentiating or integrating. The half-angle identities are especially important for integrating even powers of sin and cos.
Pythagorean identities
sin^2 x + cos^2 x = 1
1 + cot^2 x = csc^2 x
tan^2 x + 1 = sec^2 x
Double-angle identities
sin 2x = 2 sin x cos x
cos 2x = cos^2 x - sin^2 x
Half-angle (power-reducing) identities
cos^2 x = (1 + cos 2x) / 2
sin^2 x = (1 - cos 2x) / 2
These two are essential for integrating sin^2 x or cos^2 x, which cannot be integrated directly.
Sum and difference identities
cos(x + y) = cos x cos y - sin x sin y
sin(x + y) = sin x cos y + cos x sin y
Exponential derivatives
d/dx (e^x) = e^x
d/dx (b^x) = (ln b) b^x
The function e^x is the only function that is its own derivative.
Exponential integrals
integral of e^x dx = e^x + C
integral of b^x dx = b^x / ln b + C
Logarithmic derivatives
d/dx (ln|x|) = 1/x
d/dx (log_b |x|) = 1 / ((ln b) x)
Logarithmic integrals
integral of 1/x dx = ln|x| + C
This is the case the power rule misses: when n = -1, the power rule formula divides by zero, and the integral is ln|x| instead.
Definition and change of base
log_b N = x means b^x = N
Change of base formula: log_b x = ln x / ln b = log x / log b
Key identities
ln(e^x) = x and e^(ln x) = x
log_b(b^x) = x and b^(log_b x) = x
ln e = 1, log 10 = 1, log_b b = 1
ln 1 = 0, log 1 = 0, log_b 1 = 0
Formula: integral of u dv = uv - integral of v du
This is the integration counterpart of the product rule. Use it when the integrand is a product of two functions where one becomes simpler upon differentiation (that one is u) and the other is easy to integrate (that one is dv).
A common mnemonic for choosing u is LIATE: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Pick u from whichever category comes first in this list.
Derivatives measure how quickly things change: velocity is the derivative of position, and acceleration is the derivative of velocity. Engineers use the chain rule every time they model a quantity that depends on an intermediate variable, such as how air pressure changes with altitude which itself changes with time.
Integrals measure accumulated totals: the area under a velocity curve gives total distance travelled, and integrating a rate of flow gives total volume. Integration by parts shows up in signal processing, where you need to integrate products of exponentials and trigonometric functions to analyse frequencies in audio or radio signals.
Students often forget the negative sign in the derivatives of cos, cot, and csc. All three "co" functions have a negative in their derivative. If you write d/dx (cos x) = sin x without the minus sign, that is wrong.
Students often apply the power rule to e^x and write d/dx (e^x) = x e^(x-1). The power rule applies to x^n (variable base, constant exponent), not to e^x (constant base, variable exponent). The derivative of e^x is simply e^x.
Students often forget the + C on indefinite integrals. Every indefinite integral represents a family of functions differing by a constant. Leaving off + C loses marks on nearly every exam.
Students often try to integrate sin^2 x directly using the power rule. You cannot do this. You must first apply the half-angle identity sin^2 x = (1 - cos 2x)/2, then integrate term by term.
The chain rule is tested on virtually every exam, often embedded inside trig or exponential problems. If you see a composite function, you need it.
Trig integrals (especially sec x and csc x) are commonly tested. The integrals ln|sec x + tan x| and ln|csc x - cot x| are difficult to derive from scratch, so memorise them.
Inverse trig integral patterns (arcsin, arctan, arcsec forms) appear frequently in integration problems. Recognise the denominator patterns: sqrt(a^2 - u^2) signals arcsin, a^2 + u^2 signals arctan, u sqrt(u^2 - a^2) signals arcsec.
Integration by parts is typically tested with products like x e^x, x sin x, or x ln x. Be ready to apply it twice in succession for integrals like e^x sin x.
True or false: d/dx (cos x) = sin x.
False. d/dx (cos x) = -sin x.
Fill in the blank: integral of x^3 dx = ____.
x^4 / 4 + C.
True or false: d/dx (e^x) = x e^(x-1).
False. d/dx (e^x) = e^x. The power rule does not apply here.
Fill in the blank: The half-angle identity for sin^2 x is ____.
(1 - cos 2x) / 2.
True or false: integral of 1/x dx = x^0 / 0 + C.
False. The integral is ln|x| + C. The power rule does not apply when n = -1.
Q: Find d/dx [x^3 sin x].
A: Apply the product rule. d/dx [x^3 sin x] = x^3 cos x + 3x^2 sin x.
Q: Evaluate integral of sec^2 x dx.
A: tan x + C. This is a direct formula.
Q: Find d/dx [ln(x^2 + 1)].
A: Apply the chain rule. The derivative of ln(u) is 1/u, multiplied by the derivative of the inner function: 2x / (x^2 + 1).
Q: Evaluate integral of 1 / (9 + x^2) dx.
A: Recognise the arctan pattern with a = 3. The answer is (1/3) tan^-1(x/3) + C.
Q: Use integration by parts to evaluate integral of x e^x dx.
A: Let u = x, dv = e^x dx. Then du = dx, v = e^x. Apply the formula: x e^x - integral of e^x dx = x e^x - e^x + C.
These formulas connect directly to infinite series (the next major unit), where you will use derivatives and integrals to find Taylor and Maclaurin series expansions of functions. The ratio and root tests for series convergence rely on limits of ratios, which are themselves built on derivative reasoning.
Integration by parts reappears in differential equations, where it is used to solve first-order linear equations. The trig identities here are also foundational for Calc 2 topics like trigonometric substitution and partial fractions.
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