Key Terms

Key Terms

acceleration

is the rate of change of the velocity, that is, the derivative of velocity

amount of change

the amount of a function $f\left(x\right)$ over an interval $\left[x,x+h\right]$ is $f\left(x+h\right)-f\left(x\right)$

average rate of change

is a function $f\left(x\right)$ over an interval $\left[x,x+h\right]$ is $\frac{f\left(a+h\right)-f\left(a\right)}{h}$

chain rule

the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function

constant multiple rule

the derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative: $\frac{d}{dx}\left(cf\left(x\right)\right)=c{f}^{\prime }\left(x\right)$

constant rule

the derivative of a constant function is zero: $\frac{d}{dx}\left(c\right)=0,$ where c is a constant

derivative

the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative

derivative function

gives the derivative of a function at each point in the domain of the original function for which the derivative is defined

difference quotient

of a function $f\left(x\right)$ at $a$ is given by

$$\frac{f\left(a+h\right)-f(a)}{h}\text{or}\frac{f\left(x\right)-f(a)}{x-a}$$

difference rule

the derivative of the difference of a function f and a function g is the same as the difference of the derivative of f and the derivative of g: $\frac{d}{dx}\left(f\left(x\right)-g\left(x\right)\right)=f^{\prime }\left(x\right)-g^{\prime }\left(x\right)$

differentiable at a

a function for which $f^{\prime }(a)$ exists is differentiable at $a$

differentiable function

a function for which $f^{\prime }(x)$ exists is a differentiable function

differentiable on S

a function for which $f^{\prime }(x)$ exists for each $x$ in the open set $S$ is differentiable on $S$

differentiation

the process of taking a derivative

higher-order derivative

a derivative of a derivative, from the second derivative to the nth derivative, is called a higher-order derivative

implicit differentiation

is a technique for computing $\frac{dy}{dx}$ for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable $y$ as a function) and solving for $\frac{dy}{dx}$

instantaneous rate of change

the rate of change of a function at any point along the function $a,$ also called $f^{\prime }(a),$ or the derivative of the function at $a$

logarithmic differentiation

is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly

marginal cost

is the derivative of the cost function, or the approximate cost of producing one more item

marginal profit

is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item

marginal revenue

is the derivative of the revenue function, or the approximate revenue obtained by selling one more item

population growth rate

is the derivative of the population with respect to time

power rule

the derivative of a power function is a function in which the power on $x$ becomes the coefficient of the term and the power on $x$ in the derivative decreases by 1: If $n$ is an integer, then $\frac{d}{dx}{x}^n=n{x}^{n-1}$

product rule

the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: $\frac{d}{dx}\left(f\left(x\right)g(x)\right)=f^{\prime }\left(x\right)g\left(x\right)+g^{\prime }\left(x\right)f\left(x\right)$

quotient rule

the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f^{\prime }\left(x\right)g\left(x\right)-g^{\prime }\left(x\right)f\left(x\right)}{{\left(g\left(x\right)\right)}^2}$

speed

is the absolute value of velocity, that is, $\left|v\left(t\right)\right|$ is the speed of an object at time $t$ whose velocity is given by $v\left(t\right)$

sum rule

the derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g: $\frac{d}{dx}\left(f\left(x\right)+g\left(x\right)\right)=f^{\prime }\left(x\right)+g^{\prime }\left(x\right)$