Rate of Change at a Point - AP Calculus AB

$e^x$. The exponential function is its own derivative.

Linear function. Linear functions have the same rate everywhere.

1. $f'(x) = 6x + 2$, so $f'(3) = 18 + 2 = 20$.

1. Using $f'(x) = 3x^2$, so $f'(1) = 3$.

$2x + 3$. Apply power rule to each term.

Derivative. The derivative measures instantaneous rate of change.

1. Constants have no rate of change.

$\text{cos}(x)$. The derivative of sine is cosine.

$\frac{f(b) - f(a)}{b - a}$. Standard formula for average rate of change over an interval.

Slope of tangent line at $x=a$. The derivative equals the slope of the line tangent to the curve.

$nx^{n-1}$. The power rule for differentiation.

1. Using $f'(x) = 2x$, so $f'(2) = 4$.

Zero. Constants have no rate of change.

Derivative $f'(a)$. The derivative gives the instantaneous rate of change at a specific point.

1. $\frac{9-1}{3-1} = \frac{8}{2} = 4$

Slope of secant line. Connects two points on the function graph.

Slope of tangent line. The tangent touches the curve at exactly one point.

Average rate = Slope of secant line. Secant lines connect two points, giving average rate.

$\frac{f(a+h) - f(a)}{h}$. The ratio used in the limit definition of derivatives.

When $f(x)$ is linear. Linear functions have constant slope everywhere.

Velocity function. When position is $f(x)$, $f'(x)$ represents velocity.

$f'(a)$. Standard notation for the derivative at a specific point.

Slope of the tangent line. The derivative represents the slope of the tangent line.

1. Since $f'(x) = e^x$ and $e^0 = 1$.

1. $\frac{8-1}{2-1} = 7$

$-\frac{1}{x^2}$. Derivative of $x^{-1}$ using the power rule.

$\sec^2(x)$. The derivative of tangent is secant squared.

$\frac{1}{x}$. The derivative of the natural logarithm function.

1. Linear functions have constant slope of 3.