Glossary

Chain Rule

Criticality: 3

A rule used to differentiate composite functions, stating that the derivative of $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

Example:

To find the derivative of $y = \sin(x^2)$ , you'd apply the Chain Rule to get $y' = \cos(x^2) \cdot 2x$ .

Constant Multiple Rule

Criticality: 2

A differentiation rule stating that the derivative of a constant times a function is the constant times the derivative of the function.

Example:

If $f(x) = 5x^3$ , then $f'(x) = 5 \cdot (3x^2) = 15x^2$ , applying the Constant Multiple Rule.

Derivative of cos x

Criticality: 3

The rate of change of the cosine function, which is the negative of the sine function.

Example:

When modeling the oscillation of a spring with $x(t) = \cos t$ , the velocity of the spring is $v(t) = -\sin t$ , found by taking the Derivative of cos x.

Derivative of e^x

Criticality: 3

The rate of change of the natural exponential function, which is unique in that it is equal to itself.

Example:

If a population grows exponentially according to $P(t) = 100e^t$ , then the rate of population growth, $P'(t)$ , is also $100e^t$ , demonstrating the special property of the Derivative of e^x.

Derivative of ln x

Criticality: 3

The rate of change of the natural logarithm function, which is the reciprocal of x.

Example:

To find the rate at which a quantity modeled by $Q(x) = \ln x$ changes, you'd calculate $Q'(x) = \frac{1}{x}$ , using the rule for the Derivative of ln x.

Derivative of sin x

Criticality: 3

The rate of change of the sine function, which is the cosine function.

Example:

If a particle's position is given by $s(t) = \sin t$ , its velocity is $v(t) = \cos t$ , meaning its instantaneous rate of change of position is the Derivative of sin x.

Power Rule

Criticality: 3

A fundamental differentiation rule used to find the derivative of functions in the form $x^n$, where the derivative is $nx^{n-1}$.

Example:

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

Product Rule

Criticality: 2

A rule for differentiating the product of two functions, given by $(uv)' = u'v + uv'$.

Example:

If you need to find the derivative of $f(x) = x^2 \sin x$ , you would use the Product Rule to get $f'(x) = 2x \sin x + x^2 \cos x$ .

Quotient Rule

Criticality: 2

A rule for differentiating the quotient of two functions, given by $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$.

Example:

To differentiate $g(x) = \frac{e^x}{x}$ , you'd apply the Quotient Rule to find $g'(x) = \frac{e^x \cdot x - e^x \cdot 1}{x^2}$ .

Second Derivative

Criticality: 3

The derivative of the first derivative of a function, representing the rate of change of the slope of the tangent line or the concavity of the function.

Example:

If $s(t)$ is position, $s'(t)$ is velocity, and $s''(t)$ is acceleration, which is the Second Derivative of position.

Tangent Line

Criticality: 3

A straight line that touches a curve at a single point and has the same slope as the curve at that point, representing the instantaneous rate of change.

Example:

To find the equation of the Tangent Line to $f(x) = x^2$ at $x=2$ , you'd find $f(2)=4$ and $f'(2)=4$ , leading to $y-4=4(x-2)$ .

Trig Identities

Criticality: 1

Equations involving trigonometric functions that are true for all values of the variables for which the functions are defined, often used to simplify expressions before differentiation.

Example:

Before differentiating $\sin^2 x + \cos^2 x$ , you could simplify it to $1$ using a fundamental Trig Identity, making its derivative $0$ .

Glossary

Chain Rule

Criticality: 3

A rule used to differentiate composite functions, stating that the derivative of $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

Example:

To find the derivative of $y = \sin(x^2)$ , you'd apply the Chain Rule to get $y' = \cos(x^2) \cdot 2x$ .

Constant Multiple Rule

Criticality: 2

A differentiation rule stating that the derivative of a constant times a function is the constant times the derivative of the function.

Example:

If $f(x) = 5x^3$ , then $f'(x) = 5 \cdot (3x^2) = 15x^2$ , applying the Constant Multiple Rule.

Derivative of cos x

Criticality: 3

The rate of change of the cosine function, which is the negative of the sine function.

Example:

When modeling the oscillation of a spring with $x(t) = \cos t$ , the velocity of the spring is $v(t) = -\sin t$ , found by taking the Derivative of cos x.

Derivative of e^x

Criticality: 3

The rate of change of the natural exponential function, which is unique in that it is equal to itself.

Example:

If a population grows exponentially according to $P(t) = 100e^t$ , then the rate of population growth, $P'(t)$ , is also $100e^t$ , demonstrating the special property of the Derivative of e^x.

Derivative of ln x

Criticality: 3

The rate of change of the natural logarithm function, which is the reciprocal of x.

Example:

To find the rate at which a quantity modeled by $Q(x) = \ln x$ changes, you'd calculate $Q'(x) = \frac{1}{x}$ , using the rule for the Derivative of ln x.

Derivative of sin x

Criticality: 3

The rate of change of the sine function, which is the cosine function.

Example:

If a particle's position is given by $s(t) = \sin t$ , its velocity is $v(t) = \cos t$ , meaning its instantaneous rate of change of position is the Derivative of sin x.

Power Rule

Criticality: 3

A fundamental differentiation rule used to find the derivative of functions in the form $x^n$, where the derivative is $nx^{n-1}$.

Example:

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

Product Rule

Criticality: 2

A rule for differentiating the product of two functions, given by $(uv)' = u'v + uv'$.

Example:

If you need to find the derivative of $f(x) = x^2 \sin x$ , you would use the Product Rule to get $f'(x) = 2x \sin x + x^2 \cos x$ .

Quotient Rule

Criticality: 2

A rule for differentiating the quotient of two functions, given by $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$.

Example:

To differentiate $g(x) = \frac{e^x}{x}$ , you'd apply the Quotient Rule to find $g'(x) = \frac{e^x \cdot x - e^x \cdot 1}{x^2}$ .

Second Derivative

Criticality: 3

The derivative of the first derivative of a function, representing the rate of change of the slope of the tangent line or the concavity of the function.

Example:

If $s(t)$ is position, $s'(t)$ is velocity, and $s''(t)$ is acceleration, which is the Second Derivative of position.

Tangent Line

Criticality: 3

A straight line that touches a curve at a single point and has the same slope as the curve at that point, representing the instantaneous rate of change.

Example:

To find the equation of the Tangent Line to $f(x) = x^2$ at $x=2$ , you'd find $f(2)=4$ and $f'(2)=4$ , leading to $y-4=4(x-2)$ .

Trig Identities

Criticality: 1

Equations involving trigonometric functions that are true for all values of the variables for which the functions are defined, often used to simplify expressions before differentiation.

Example:

Before differentiating $\sin^2 x + \cos^2 x$ , you could simplify it to $1$ using a fundamental Trig Identity, making its derivative $0$ .