Derivative of as a Function
If a function f(x) is said to be differentiable at a, then this must exist.
What is f'(a)
This is the name given to the expression \frac{f(x) - f(a)}{x - a}
What is the difference quotient
The derivative of a constant function $f(x) = c$ is equal to this
What is zero
the derivative of sin(x) with respect to x is this
What is cos(x)
If s(t) is the position of an object, then its rate of change with respect to time, s'(t), represents this
What is velocity
The function whose domain consists of all values of x where the limit \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} exists is called this
What is the derivative function (denoted by f'(x))
The slope of a line connecting two points on a curve is called this
What is the secant line
This rule states that the derivative of x^n, where n is a positive integer, is nx^{n-1}
What is the power rule
The derivative of cos(x) with respect to x is this
The magnitude (absolute value) of the velocity of an object is defined as this
What is speed
If a function f(x) is differentiable at a point a, then it must also be this at a
What is continuous
The derivative of a function at a point is defined as the limit of this as x approaches that point
What is the slope of the secant line or difference quotient
the derivative of the sum of two differentiable functions f(x) and g(x) is equal to this
The derivative of tan(x) with respect to x is this
The rate of change of velocity with respect to time is known as this
What is acceleration
A function may fail to be differentiable at a point where its graph has one of these, such as in the case of f(x) = |x| at x=0
What is sharp corner
This line to f(x) at a passes through the point (a, f(a)) and has a slope equal to \lim_{x \to a} \frac{f(x) - f(a)}{x - a}, provided this limit exists
What is the tangent line
This rule is used to find the derivative of the product of two differentiable functions f(x) and g(x) and is given by f'(x)g(x) + g'(x)f(x)
What is the product rule
The derivative of the natural exponential function f(x) = e^x is this
What is e^x
In economics, if C(x) is the cost of producing x items, then C'(x) is known as this
What is marginal cost
A function may also fail to be differentiable at a point where there is one of these on its graph, as seen in the example of f(x) = \sqrt{x^3} at x=0
What is vertical tangent line
The instantaneous rate of change of a position function $s(t)$ at a specific time a is known as this
What is the instantaneous velocity
To find the derivative of a composite function f(g(x)), this rule states that the derivative is f'(g(x))g'(x)
What is the chain rule
The derivative of the natural logarithmic function y = ln(x) is this
What is frac{1}{x}
According to the amount of change formula, a new value f(a+h) can be estimated using the old value f(a), the derivative f'(a), and a small change $h$ with this approximation
What is f(a+h) \approx f(a) + f'(a)h