Mastering Basic Derivative Techniques

Differentiation

The algebraic process of finding the derivative, representing the instantaneous rate of change.

Power Rule

For any real number n, the derivative of x^n is nx^(n-1).

Constant Rule

The derivative of a constant c is zero.

Constant Multiple Rule

The derivative of c * f(x) is c * f'(x).

Sum and Difference Rule

The derivative of f(x) ± g(x) is f'(x) ± g'(x).

Trigonometric Function Derivative: sin(x)

The derivative of sin(x) is cos(x).

Trigonometric Function Derivative: cos(x)

The derivative of cos(x) is -sin(x).

Trigonometric Function Derivative: tan(x)

The derivative of tan(x) is sec^2(x).

Natural Base e

The derivative of e^x is e^x.

Logarithmic Derivative

The derivative of ln(x) is 1/x.

Product Rule

For f(x)g(x), the derivative is f(x)g'(x) + g(x)f'(x).

Quotient Rule

For f(x)/g(x), the derivative is (g(x)f'(x) - f(x)g'(x)) / [g(x)]^2.

Critical Pre-Step

Rewriting functions into the form x^n before applying the Power Rule.

Rational Functions

Functions involving fractions, which can often be rewritten to apply the Power Rule.

Chain Rule

A method for differentiating composite functions, not listed in basic rules.

Differentiation Operator

An operator that applies to calculus functions, following specific rules for calculation.

Derivative of a Constant

The slope (derivative) of a horizontal line is zero.

Coefficients in Derivatives

When differentiating a function with a coefficient, maintain the coefficient in derivative calculations.

Instantaneous Rate of Change

The value of the derivative at a specific point, reflecting the slope of the tangent line.

Algebra Errors in Rewriting

Common mistakes that occur when transforming functions into power form.

Negative Exponents

Exponents that result in reciprocal values, which can be applied in the Power Rule.

Derivative of Exponential Function

For a base a (where a > 0 and a ≠ 1), the derivative is a^x ln(a).

Derivative of Cosecant

The derivative of csc(x) is -csc(x)cot(x).

Derivative of Cotangent

The derivative of cot(x) is -csc^2(x).

Derivative of Secant

The derivative of sec(x) is sec(x)tan(x).

Common Mistakes in Calculus

Frequent errors found in differentiation, particularly with rules and order of operations.

Algebraic Expressions

Mathematical phrases including constants, variables, and operators, used in differentiation.