The rate at which a function's output changes with respect to its input at a specific point. It's the derivative.

Derivatives don't exist at sharp corners, cusps, vertical tangents, or points of discontinuity.

A positive derivative indicates the function is increasing, a negative derivative indicates it is decreasing, and a zero derivative indicates a stationary point.

If a function is differentiable at a point, it must be continuous at that point. However, a continuous function is not necessarily differentiable.

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

Find the derivative at the given point to get the slope. Use the point-slope form of a line: $y - y_1 = m(x - x_1)$.

Derivatives can be used to find maximum and minimum values of a function by finding critical points (where the derivative is zero or undefined).

Higher-order derivatives are derivatives of derivatives. For example, the second derivative is the derivative of the first derivative, and so on.

The second derivative tells you about the concavity of a function. A positive second derivative indicates concave up, and a negative second derivative indicates concave down.

Average rate of change is over an interval, while instantaneous rate of change is at a single point.

Apply the power rule to each term: multiply by the exponent and reduce the exponent by 1. Use the sum/difference rule for multiple terms.

Use the product rule: (first function) * (derivative of second) + (second function) * (derivative of first).

Use the quotient rule: [(bottom) * (derivative of top) - (top) * (derivative of bottom)] / (bottom)^2.

Use the chain rule: $f'(x) = ke^{kx}$.

Use the chain rule: $f'(x) = \frac{g'(x)}{g(x)}$.

1. Find $f(a)$. 2. Find $f'(x)$. 3. Find $f'(a)$ (the slope). 4. Use point-slope form: $y - f(a) = f'(a)(x - a)$.

1. Find $f'(x)$. 2. Find critical points where $f'(x) = 0$ or is undefined. 3. Test intervals around critical points to determine the sign of $f'(x)$.

1. Find critical points. 2. Use the first derivative test (sign change of $f'(x)$) or the second derivative test (sign of $f''(x)$).

1. Find $f''(x)$. 2. Find points where $f''(x) = 0$ or is undefined. 3. Test intervals around these points to determine the sign of $f''(x)$.

1. Find $f''(x)$. 2. Find points where $f''(x) = 0$ or is undefined. 3. Check that the concavity changes at these points.

The instantaneous rate of change of a function.

The derivative of *y* with respect to *x*, indicating the rate of change of *y* with respect to *x*.

A line that touches a curve at only one point (locally) and has the same slope as the curve at that point.

A point where a curve has a sharp point and the derivative is undefined.

The speed of an object at a specific moment in time; represented by the derivative of the position function.

$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

The derivative of a function at a point is equal to the slope of the tangent line at that point.

A tangent line that is vertical, indicating the derivative is undefined at that point.

A function is differentiable at a point if its derivative exists at that point.

The change in the value of a function divided by the change in the independent variable over a given interval.