What is a derivative?
The instantaneous rate of change of a function.
What does $\frac{dy}{dx}$ represent?
The derivative of *y* with respect to *x*, indicating the rate of change of *y* with respect to *x*.
What is a tangent line?
A line that touches a curve at only one point (locally) and has the same slope as the curve at that point.
What is a cusp?
A point where a curve has a sharp point and the derivative is undefined.
What is instantaneous speed?
The speed of an object at a specific moment in time; represented by the derivative of the position function.
What is the formal definition of the derivative of f(x)?
$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
What is the relationship between a derivative and a tangent line?
The derivative of a function at a point is equal to the slope of the tangent line at that point.
What is a vertical tangent?
A tangent line that is vertical, indicating the derivative is undefined at that point.
What does it mean for a function to be differentiable?
A function is differentiable at a point if its derivative exists at that point.
What is the average rate of change?
The change in the value of a function divided by the change in the independent variable over a given interval.
What is the power rule?
If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
What is the product rule?
If $h(x) = f(x)g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$.
What is the quotient rule?
If $h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.
What is the derivative of sin(x)?
$\frac{d}{dx} \sin(x) = \cos(x)$
What is the derivative of cos(x)?
$\frac{d}{dx} \cos(x) = -\sin(x)$
What is the derivative of tan(x)?
$\frac{d}{dx} \tan(x) = \sec^2(x)$
What is the derivative of $e^x$?
$\frac{d}{dx} e^x = e^x$
What is the derivative of ln(x)?
$\frac{d}{dx} \ln(x) = \frac{1}{x}$
What is the constant rule?
If $f(x) = c$, where c is a constant, then $f'(x) = 0$.
What is the constant multiple rule?
$\frac{d}{dx}[cf(x)] = c \cdot f'(x)$
What is the sum rule?
$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$
Explain the concept of instantaneous rate of change.
The rate at which a function's output changes with respect to its input at a specific point. It's the derivative.
Explain when a derivative does not exist.
Derivatives don't exist at sharp corners, cusps, vertical tangents, or points of discontinuity.
What does the sign of the derivative tell you?
A positive derivative indicates the function is increasing, a negative derivative indicates it is decreasing, and a zero derivative indicates a stationary point.
Explain the relationship between differentiability and continuity.
If a function is differentiable at a point, it must be continuous at that point. However, a continuous function is not necessarily differentiable.
What is the geometric interpretation of the derivative?
The derivative at a point represents the slope of the tangent line to the curve at that point.
Explain how to find the equation of a tangent line.
Find the derivative at the given point to get the slope. Use the point-slope form of a line: $y - y_1 = m(x - x_1)$.
How can derivatives be used in optimization problems?
Derivatives can be used to find maximum and minimum values of a function by finding critical points (where the derivative is zero or undefined).
Explain the concept of higher-order derivatives.
Higher-order derivatives are derivatives of derivatives. For example, the second derivative is the derivative of the first derivative, and so on.
What does the second derivative tell you about a function?
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.
Explain the difference between average and instantaneous rate of change.
Average rate of change is over an interval, while instantaneous rate of change is at a single point.
