Calculus learning library

Learn calculus rules from the reason, not just the answer

Read concise proofs, domain conditions, worked examples and common mistakes, then try the same problem in the calculator.

Function derivative formulas and methods

Exponential derivatives

Derivative of exp(x): Why eˣ Is Its Own Derivative

$$\frac{\mathrm{d}}{\mathrm{d}x}\left[e^{x}\right] = e^{x}$$

For a real aˣ, a>0; exp(g(x)) also requires g′(x).

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Core derivative rules

Power Rule for Derivatives: Exponents and Conditions

d/dx xⁿ = nxⁿ⁻¹

The real interval must support the original power; a composite base also requires the chain rule.

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Differentiation methods

How to Use the Product Rule Without Losing a Term

(fg)′ = f′g + fg′

Both factors may depend on x; a true constant factor can be pulled out.

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Differentiation methods

How to Use the Quotient Rule and Keep the Order

(f/g)′ = (f′g − fg′)/g²

The denominator g(x) must be nonzero.

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Higher derivatives

How to Find a Second Derivative Step by Step

f″(x) = d/dx [f′(x)]

The first derivative must itself be differentiable on the interval.

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Radical derivatives

Derivative of √x: Formula, Domain, and x=0

d/dx √x = 1/(2√x)

Over the reals √x exists for x≥0, but this finite derivative requires x>0.

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Piecewise derivatives

Derivative of |x|: Piecewise Rule and the Corner at Zero

d/dx |x| = x/|x| for x≠0; undefined at x=0

The formula excludes zero; every zero of a composite inside expression needs direct analysis.

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Trigonometric derivatives

Derivative of sin(x): Why It Is cos(x)

$$\frac{\mathrm{d}}{\mathrm{d}x}\left[\sin x\right] = \cos x$$

The standard formula assumes radians; sin(g(x)) also requires multiplication by g′(x).

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Trigonometric derivatives

Derivative of cos(x): The Negative Sine Sign Explained

$$\frac{\mathrm{d}}{\mathrm{d}x}\left[\cos x\right] = -\sin x$$

The standard formula assumes radians; cos(g(x)) also requires g′(x).

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Reciprocal derivatives

Derivative of 1/x: Power Rule, Chain Rule and Domain

$$\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{1}{x}\right] = \frac{-1}{x^{2}}$$

The function and derivative are defined only for x ≠ 0. For 1/g(x), require g(x) ≠ 0 and differentiability.

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Natural logarithms

Derivative of ln(x): Formula, Proof, and Examples

$$\frac{\mathrm{d}}{\mathrm{d}x}\left[\ln x\right] = \frac{1}{x}$$

For real-valued ln(x), x > 0.

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Inverse trigonometric functions

Derivative of arctan(x): Formula, Proof, and Examples

$$\frac{\mathrm{d}}{\mathrm{d}x}\left[\arctan x\right] = \frac{1}{1 + x^{2}}$$

Valid for every real x; angles are measured in radians.

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Trigonometric derivatives

Derivative of tan(x): Formula, Proof, and Examples

$$\frac{\mathrm{d}}{\mathrm{d}x}\left[\tan x\right] = \sec^{2} x$$

Requires cos(x) ≠ 0; angles are measured in radians.

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Reciprocal trigonometric functions

Derivative of sec(x): Formula, Proof, and Examples

$$\frac{\mathrm{d}}{\mathrm{d}x}\left[\sec x\right] = \sec x\,\tan x$$

Requires cos(x) ≠ 0; angles are measured in radians.

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Differentiation methods

How to Find Partial Derivatives: Steps and Examples

Choose one variable; hold the other independent variables constant.

A partial derivative describes change in one independent direction while the remaining independent variables stay fixed.

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Differentiation methods

How to Use the Chain Rule: Steps and Examples

If y=f(g(x)), then y′=f′(g(x))g′(x).

Use the chain rule when one differentiable function is evaluated inside another differentiable function.

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Differentiation methods

How to Do Implicit Differentiation: Steps and Examples

Differentiate both sides, collect every y′ term, then solve for y′.

A local slope formula for y=y(x) requires the coefficient of y′ to be nonzero at the point being studied.

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Chain-rule mistakes

Why Do You Multiply by the Inner Derivative?

For y=f(g(x)), multiply by g'(x) because the input of f changes at the rate g'(x), not at one unit per x.

The rule applies when one differentiable function is evaluated inside another; every nontrivial nested layer contributes a factor.

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Choosing a derivative rule

Chain Rule vs. Product Rule: How to Tell the Difference

Use the chain rule for f(g(x)), the product rule for f(x)g(x), and apply both when a multiplied factor contains a composition.

Identify the top-level operation before differentiating: function application means composition, while an explicit product joins separate factors.

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Partial-derivative meaning

Why Are Other Variables Constant in Partial Derivatives?

A partial derivative measures change in one coordinate direction, so the other independent variables are held fixed during that measurement.

This interpretation assumes the listed variables are independent coordinates; variables connected by a constraint require a different analysis.

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Choosing the active variable

Partial Derivative with Respect to x vs. y

With respect to x, hold y fixed; with respect to y, hold x fixed. The same expression can therefore produce two different valid derivatives.

Always identify the active variable before applying derivative rules, and preserve any other variables that act as coefficients.

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Integration methods and definite integrals

Integration methods

Integration by Substitution: Match the Inner Derivative

If u=g(x), then ∫f(g(x))g′(x)dx=∫f(u)du

The substitution must account for the differential factor and preserve or transform the bounds.

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Integration methods

Integration by Parts: Choosing u and dv

$$\int u\,\mathrm{d}v$$

Choose dv so it can be integrated and u so differentiation makes the remaining integral simpler.

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Logarithmic integrals

Integral of 1/x: Why It Is ln|x| + C

$$\int \frac{1}{x}\,\mathrm{d}x = \ln\left|x\right| + C$$

The integrand is defined only for x≠0, so antiderivatives live on intervals that do not cross zero.

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Trigonometric integrals

Integrals of sin(x) and cos(x): Signs and Factors

∫sin(x)dx=−cos(x)+C; ∫cos(x)dx=sin(x)+C

For sin(ax+b) or cos(ax+b), divide by the nonzero inner slope a.

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Definite integral methods

How to Evaluate Definite Integrals: Bounds and Worked Steps

$$\int_{\ldots}^{\ldots} f\,x\,\mathrm{d}x = F\,b - F\,a$$

This form of the fundamental theorem applies when f is continuous on the closed finite interval and F′=f. Singularities and improper integrals need separate analysis.

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Integration methods

U-Substitution vs. Integration by Parts: How to Choose

Use substitution for a composite pattern f(g(x))g′(x); use integration by parts when a product becomes simpler after differentiating one factor.

This is a method-selection test, not a guarantee that either technique finishes every integral.

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Definite integrals

Why a Definite Integral Is Not Always the Total Area

A definite integral gives net signed area; total area adds the magnitudes of regions above and below the axis.

If the integrand changes sign on the interval, split at every relevant zero before calculating total area.

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Limits and directional behavior

Algebraic limits

Limits by Factoring: Remove a Hole Correctly

Factor, cancel the common nonzero factor near the point, then evaluate.

Cancellation is valid only for nearby x where the canceled factor is nonzero.

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Directional limits

One-Sided Limits: Read Left and Right Separately

A two-sided limit exists only when the left and right limits agree.

The function value at the target does not determine either one-sided limit.

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Limit methods

When to Use L’Hôpital’s Rule—and When Not To

Use L’Hôpital only after substitution gives 0/0 or ∞/∞ and its hypotheses hold.

The quotient must have an eligible indeterminate form, and the differentiated quotient must be meaningful near the target.

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One-sided limits

When Left and Right Limits Differ, the Limit Does Not Exist

A two-sided limit exists only when the left-hand and right-hand limits approach the same value.

Check both directions separately whenever the expression changes behavior across the approach point.

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