Derivatives are the key to understanding how things change. They help us calculate rates of change, like velocity and acceleration, which are crucial in physics and engineering. We use them to analyze motion and predict how objects move.

Differentiation rules are the tools we use to find derivatives. From the basic power rule to the more complex chain rule, these techniques let us tackle a wide range of functions and solve real-world problems involving motion and change.

Rates of Change and Motion

Derivative as rate of change

• Represents instantaneous rate of change measures how quickly a function changes at a specific point
• Velocity rate of change of position with respect to time derivative of position function gives velocity (distance traveled over time)
• Acceleration rate of change of velocity with respect to time derivative of velocity function gives acceleration (change in speed over time)

Applications of differentiation rules

• Power rule find derivatives of polynomial functions $\frac{d}{dx}(x^n) = nx^{n-1}$ (square function $x^2$, cube function $x^3$)
• Constant multiple rule differentiate functions with coefficients $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}f(x)$, $c$ is a constant (2x, 3x)
• Sum and difference rules differentiate combinations of functions $\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)$ (polynomials, trigonometric functions)
• Product rule more complex functions $\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x)$ (polynomial multiplied by exponential function)
• Quotient rule more complex functions $\frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{[g(x)]^2}$ (rational functions)
• Chain rule composite functions $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$ (function inside another function like $\sin(x^2)$)

Position, velocity, and acceleration relationships

• Position location of an object at a given time (distance from origin)
• Velocity first derivative of position with respect to time represents rate of change of position indicates speed and direction of motion
• Acceleration second derivative of position or first derivative of velocity with respect to time represents rate of change of velocity indicates how quickly velocity changes

Instantaneous motion from position functions

• Given position function $s(t)$, find velocity by taking first derivative velocity function: $v(t) = \frac{d}{dt}s(t)$
• Find acceleration by taking second derivative of position or first derivative of velocity acceleration function: $a(t) = \frac{d^2}{dt^2}s(t) = \frac{d}{dt}v(t)$
• Evaluate velocity and acceleration functions at specific time to determine instantaneous values (velocity at t=2 seconds, acceleration at t=3 seconds)

Motion analysis with derivatives

1. Identify given information and desired quantity to solve for (initial position, velocity, time)
2. Set up position function based on problem description (quadratic for constant acceleration, trigonometric for periodic motion)
3. Use differentiation rules to find velocity and acceleration functions
4. Evaluate functions at specific times or solve for unknown variables (position at t=5 seconds, time when velocity is zero)
5. Interpret results in context of problem
   • Positive velocity indicates motion in positive direction, negative velocity indicates motion in negative direction
   • Positive acceleration represents increasing velocity, negative acceleration represents decreasing velocity