Derivatives and Antiderivatives

These flashcards summarize key concepts about derivatives and antiderivatives, which are crucial for understanding calculus.

Derivative

A derivative represents the rate at which a function is changing at any given point.

Antiderivative

An antiderivative is a function whose derivative is the given function.

Power Rule for Derivatives

For a function of the form $u^n$, the derivative is given by $d(u^n)/dx = nu^{n-1}$.

Sum Rule for Derivatives

The derivative of a sum of two functions is the sum of their derivatives.

Product Rule

For two functions $u$ and $v$, the derivative is given by $d(uv)/dx = u'v + uv'$.

Quotient Rule

For two functions $u$ and $v$, the derivative is given by $d(u/v)/dx = (u'v - uv')/v^2$.

Chain Rule

The derivative of a composite function is given by $d(f(g(x)))/dx = f'(g(x))g'(x)$.

Integral

An integral is a mathematical object that represents the area under a curve.

Definite Integral

A definite integral calculates the integral of a function between specific limits.

Indefinite Integral

An indefinite integral represents a family of functions and includes a constant of integration.

Integration by Parts

A technique for integrating a product of two functions, given by $\int u \, dv = uv - \int v \, du$.

Trigonometric Integrals

Integrals involving functions such as sine, cosine, secant, and cosecant.

Logarithmic Differentiation

A technique that involves taking the logarithm of both sides of an equation to simplify differentiation.