Glossary

Chain Rule

Criticality: 3

A fundamental rule for differentiating composite functions, stating that the derivative of the 'outside' function is multiplied by the derivative of the 'inside' function. If h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).

Example:

To find the derivative of $f(x) = \cos(x^2+1)$ , you apply the Chain Rule, differentiating cosine first, then the argument $x^2+1$ .

Constant Multiple Rule

Criticality: 3

States that the derivative of a constant times a function is the constant times the derivative of the function. If f(x) = c * g(x), then f'(x) = c * g'(x).

Example:

If a particle's position is given by $s(t) = 7t^2$ , its velocity is found using the Constant Multiple Rule as $v(t) = 14t$ .

Derivative of a^x

Criticality: 2

The rate of change of an exponential function with a base 'a' (where a > 0 and a ≠ 1). The derivative of a^x is a^x ln(a).

Example:

To find the rate of decay of a radioactive substance modeled by $N(t) = N_0 (0.5)^t$ , you'd use the derivative of a^x rule.

Derivative of cos(x)

Criticality: 3

The rate of change of the cosine function with respect to its input variable. The derivative of cos(x) is -sin(x).

Example:

If the temperature in a room oscillates as $T(t) = 20 + 5\cos(t)$ , the rate of temperature change is found using the derivative of cos(x), resulting in $-5\sin(t)$ .

Derivative of cot(x)

Criticality: 2

The rate of change of the cotangent function with respect to its input variable. The derivative of cot(x) is -csc^2(x).

Example:

The derivative of cot(x), $-\csc^2(x)$ , is often used when simplifying expressions or solving related rates problems involving angles.

Derivative of csc(x)

Criticality: 1

The rate of change of the cosecant function with respect to its input variable. The derivative of csc(x) is -csc(x)cot(x).

Example:

While less common in direct applications, understanding the derivative of csc(x) is crucial for differentiating complex trigonometric expressions involving reciprocals.

Derivative of e^x

Criticality: 3

The rate of change of the natural exponential function, which is unique in that its derivative is itself. The derivative of e^x is e^x.

Example:

If a population grows exponentially according to $P(t) = 100e^t$ , the rate of population growth is also $100e^t$ , thanks to the derivative of e^x.

Derivative of ln(x)

Criticality: 3

The rate of change of the natural logarithm function. The derivative of ln(x) is 1/x.

Example:

If the intensity of a sound is measured on a logarithmic scale, its rate of change with respect to some variable might involve the derivative of ln(x), which is $1/x$ .

Derivative of log_a(x)

Criticality: 1

The rate of change of a logarithm function with an arbitrary base 'a'. The derivative of log_a(x) is 1/(x ln(a)).

Example:

When working with logarithmic scales in chemistry or physics, the derivative of log_a(x) helps determine rates of change for non-natural log functions.

Derivative of sec(x)

Criticality: 1

The rate of change of the secant function with respect to its input variable. The derivative of sec(x) is sec(x)tan(x).

Example:

Similar to cosecant, knowing the derivative of sec(x) is essential for comprehensive mastery of trigonometric derivatives.

Derivative of sin(x)

Criticality: 3

The rate of change of the sine function with respect to its input variable. The derivative of sin(x) is cos(x).

Example:

If a pendulum's displacement is modeled by $y = \sin(t)$ , its velocity is given by the derivative of sin(x), which is $\cos(t)$ .

Derivative of tan(x)

Criticality: 2

The rate of change of the tangent function with respect to its input variable. The derivative of tan(x) is sec^2(x).

Example:

When analyzing the slope of a line whose angle is changing, if the slope is $\tan(\theta)$ , its rate of change with respect to $\theta$ is $\sec^2(\theta)$ , derived from the derivative of tan(x).

Derivatives of Inverse Functions

Criticality: 2

A formula that relates the derivative of an inverse function to the derivative of the original function. If y = f^(-1)(x), then dy/dx = 1 / f'(f^(-1)(x)).

Example:

If you know the derivative of $f(x) = x^3$ , you can use the formula for Derivatives of Inverse Functions to find the derivative of its inverse, $f^{-1}(x) = \sqrt[3]{x}$ , without directly differentiating $\sqrt[3]{x}$ .

Implicit Differentiation

Criticality: 3

A technique used to find the derivative of a dependent variable with respect to an independent variable when the relationship between them is not explicitly defined, often when y cannot be easily isolated.

Example:

To find $\frac{dy}{dx}$ for a circle defined by $x^2 + y^2 = 9$ , you use Implicit Differentiation because y is not easily isolated.

Power Rule

Criticality: 3

A fundamental rule for differentiating functions of the form x^n, where the exponent n is a real number. It states that if f(x) = x^n, then f'(x) = nx^(n-1).

Example:

To find the rate of change of the volume of a sphere with respect to its radius, $V = \frac{4}{3}\pi r^3$ , you'd use the Power Rule to get $V' = 4\pi r^2$ .

Product Rule

Criticality: 3

A rule used to find the derivative of a function that is the product of two differentiable functions. If h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x).

Example:

To find the derivative of $f(x) = x^2 \sin(x)$ , you must apply the Product Rule, treating $x^2$ as the first function and $\sin(x)$ as the second.

Quotient Rule

Criticality: 3

A rule used to find the derivative of a function that is the ratio (quotient) of two differentiable functions. If h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))^2.

Example:

When differentiating a rational function like $y = \frac{e^x}{x^3}$ , the Quotient Rule is indispensable.

Sum/Difference Rule

Criticality: 3

Allows you to differentiate a sum or difference of functions by differentiating each term separately and then adding or subtracting the results. If h(x) = f(x) ± g(x), then h'(x) = f'(x) ± g'(x).

Example:

When finding the derivative of a polynomial like $P(x) = 3x^4 - 2x^2 + 5x$ , you apply the Sum/Difference Rule to each term.