It's a measure of the total change over the entire domain of the function.

The derivative of a function represents the rate at which the function's value changes as its input changes. It's the limit of the difference quotient as the interval approaches zero.

The derivative is the maximum value of a function over a given interval.

The derivative is the sum of all the values a function takes over its domain.

f(x)^2

f''(x)

f(x)'-dx

f'(x) or df/dx

If f(x) = x^n, then f'(x) = nx^(n-1). This is known as the Power Rule.

f'(x) = 1/f(x)

If f(x) = nx^n, then f'(x) = x^(n-1).

The Power Rule states: f'(x) = x^(n+1).

The derivative oscillates depending on the constant.

The derivative is equal to the constant.

The derivative is equal to one.

The derivative of a constant function is always zero, because constants do not change.

The Chain Rule is used to differentiate compositions of functions. If h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).

The Chain Rule only works for polynomials.

The Chain Rule states that you add the derivatives of each component.

It's where you take the derivative of the sum of two functions.

A critical point of a function is where its derivative is zero or undefined, and it's often where a function changes from increasing to decreasing or vice versa.

A critical point is any point where the function's value is undefined.

It's a point on a curve where the slope is maximal.

A critical point is where the second derivative is zero.

The derivative provides information on the slope of the tangent line to the curve of the function at any given point.

The derivative gives the average value of the function over an interval.

It indicates the highest point on the graph.

It shows the total area under the curve.

By finding where the derivative is zero (critical points) and using the Second Derivative Test to determine if those points are maxima or minima.

The derivative only finds the minimum, not the maximum.

By finding the points where the function oscillates.

By examining where the first derivative is undefined.

The Second Derivative Test is used to determine the concavity of a function and whether a critical point is a local maximum, minimum, or saddle point.

It determines if a function has periodic behavior.

The test determines the symmetry of a function.

It computes the second integral of the derivative.

By dividing the derivative of one by the other.

By taking the square of the derivatives of each function.

Using the Product Rule: if u(x) and v(x) are functions, then the derivative of their product is u'(x)v(x) + u(x)v'(x).

By adding the derivatives of each function.

The Quotient Rule is used to differentiate the quotient of two functions: if f(x) = g(x)/h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x)) / [h(x)]^2.

The Quotient Rule only applies to polynomials.

It's the rule that states the quotient is equal to the product of derivatives.

It's the second derivative of the division of functions.

It's the process of finding antiderivatives.

Implicit differentiation is used to find the derivative of functions that are not explicitly solved for one variable.

It's the reverse of explicit differentiation.

The process of taking derivatives of constant functions.

It measures the time taken for a task.

Differentiation is used to compute rates of change, such as velocity and acceleration, and to find optimal solutions in various fields like economics and engineering.

Differentiation allows for the integration of complex functions.

It's mainly used for finding the area of shapes.

At an inflection point, the second derivative changes sign, indicating a change in the direction of concavity of the function.

The first derivative equals zero at an inflection point.

The derivative becomes undefined at an inflection point.

The point has maximum slope.

All functions can be differentiated with enough manipulation.

Not all functions can be differentiated. A function can be differentiated only if it is continuous and smooth (without sharp corners) at the point in question.

Differentiability is never restricted by the function type.

Functions are only differentiable when they have integer outputs.