The derivative of a sum of functions is the sum of their derivatives.

The derivative of a function is a measure of how the function value changes as its input changes. It represents an instantaneous rate of change and is the slope of the tangent line to the function at a point.

The derivative provides the acceleration of a function at any given point.

The derivative of a function indicates its average rate of change over an interval.

The derivative of a function f(x) can be denoted as f'(x), df/dx, or Df(x).

f(x), gf(x), or hf(x).

f'(x), ff(x), or gf(x).

df/dt, g'(x), or Df(x).

d/dx [x^n] = x^(n-2) for any real number n.

d/dx [x^n] = (n+1)x^n for any real number n.

The power rule states that d/dx [x^n] = nx^(n-1) for any real number n.

d/dx [x^n] = nx^n for any real number n.

The derivative is equal to the constant itself.

The derivative of a constant function is 0, as constant functions do not change.

The derivative of a constant function is 1.

The derivative of a constant function is infinite.

By using the chain rule: (uv)' = (du/dx) * v.

By simply adding the derivatives of both functions.

The derivative of a product of two functions, u(x) and v(x), is given by the product rule: (uv)' = u'v + uv'.

By multiplying the derivatives of the individual functions.

The chain rule states that if a variable z depends on y, which depends on x, then the derivative of z with respect to x is dz/dx = (dz/dy) * (dy/dx).

The chain rule is a rule stating to simply multiply the derivatives together if they form a chain.

The chain rule states that the derivative of a function is the derivative of its output.

The chain rule states if a variable x depends on y, then dy/dx equals x.

The derivative of a sum of functions is the product of their derivatives.

The derivative of a sum is calculated using the quotient rule.

The derivative of a sum of functions is calculated by subtracting the derivatives.

The derivative of a sum of functions is the sum of their derivatives: d/dx [u(x) + v(x)] = u'(x) + v'(x).

The derivative only provides information about the y-intercept.

The derivative gives no significant information about the graph.

The derivative tells you the x-coordinates where the function equals zero.

The derivative provides information on the slope or steepness of the graph at any given point, indicating whether the function is increasing or decreasing.

The derivative at a point represents the slope of the tangent line to the curve of the function at that point.

The derivative at a point indicates where the function crosses the y-axis at that point.

The derivative at a point describes the curvature of the entire graph.

The derivative at a point provides the area under the curve at that point.

The derivative of a quotient u(x)/v(x) is given by the quotient rule: (u/v)' = (u'v - uv')/v^2.

The derivative of a quotient is the division of their derivatives.

The derivative of a quotient is given by the product rule.

The derivative of a quotient involves simply multiplying numerator and denominator derivatives.

Implicit differentiation is simply finding the direct derivative of x.

Implicit differentiation is used to find the derivative of a function expressed in terms of two or more variables, without explicitly solving for one variable in terms of the others.

Implicit differentiation is when you find the derivative directly from the graph.

Implicit differentiation involves translating variable functions into single-variable derivatives directly.

Higher-order derivatives are only used in solving polynomial equations.

Higher-order derivatives are used primarily during company market evaluations.

Higher-order derivatives are only crucial in integral calculus.

Higher-order derivatives are used when examining the rate of change of the rate of change, often for studying concavity and the behavior of functions beyond just their slope.

The second derivative always equals the function's x-intercept.

The second derivative measures the stability of a function at a specific point.

The second derivative measures the rate of change of the first derivative, providing information on the concavity of the original function.

The second derivative is the square of the first derivative.

It is not used in determining concavity of any function.

If the second derivative of a function is positive, the function is concave up. If it is negative, the function is concave down.

Concavity depends on the value of the first derivative rather than the second derivative.

The second derivative indicates only points of intersection with the y-axis, not concavity.

The derivative calibrates non-related functions to find extrema.

The derivative helps find local maxima and minima by identifying critical points where the first derivative is zero or undefined, and then using the second derivative test to determine the nature of these points.

Derivatives are not used to find local extrema; this is done using integration.

The derivative values show only inflection points.