Calculus: Introduction to Derivatives and Differentiation Rules

Vocabulary terms and fundamental differentiation rules covered in the calculus lesson, including basic power, product, quotient, and trigonometric rules, along with geometric interpretations.

Derivative

A function that gives the slope of the tangent line to a curve at some $x$ value.

Derivative of a Constant

A differentiation rule stating that the derivative of any constant is equal to $0$, because the graph of a constant is a horizontal line with a slope of zero.

$$\frac{d}{dx}$$

A mathematical differential operator notation indicating that you are about to differentiate an expression with respect to $x$.

Power Rule

A formula used to differentiate a variable raised to a constant exponent: $\frac{d}{dx}(x^n) = n \times x^{n-1}$.

Constant Multiple Rule

A differentiation rule stating that the derivative of a constant multiplied by a function equals the constant multiplied by the derivative of that function: $\frac{d}{dx}(c \cdot f(x)) = c \cdot f'(x)$.

Definition of the Derivative

The formula used to find the derivative via the limit process: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.

Tangent Line

A line that touches a curve at only one point, whose slope is equal to the derivative evaluated at that point.

Secant Line

A line that touches or crosses a curve at two points, whose slope is found using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Derivative of $\sin(x)$

The basic trigonometric derivative equal to $\cos(x)$.

Derivative of $\cos(x)$

The trigonometric derivative equal to $-\sin(x)$.

Derivative of $\tan(x)$

The trigonometric derivative equal to $\sec^2(x)$.

Derivative of $\cot(x)$

The trigonometric derivative equal to $-\csc^2(x)$.

Derivative of $\sec(x)$

The trigonometric derivative equal to $\sec(x)\tan(x)$.

Derivative of $\csc(x)$

The trigonometric derivative equal to $-\csc(x)\cot(x)$.

Product Rule

A rule for finding the derivative of two multiplied functions: $\frac{d}{dx}(f \cdot g) = f' \cdot g + f \cdot g'$.

Three-Function Product Rule

The extension of the product rule for three multiplied functions: $\frac{d}{dx}(f \cdot g \cdot h) = f' \cdot g \cdot h + f \cdot g' \cdot h + f \cdot g \cdot h'$.

Quotient Rule

A formula used to differentiate the quotient of two functions: $\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{g \cdot f' - f \cdot g'}{g^2}$.