Derivatives are the key to understanding how things change. They help us calculate rates of change, like and , which are crucial in physics and engineering. We use them to analyze motion and predict how objects move.
are the tools we use to find derivatives. From the basic to the more complex , 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
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Represents measures how quickly a function changes at a specific point
Velocity rate of change of position with respect to time 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 dxd(xn)=nxn−1 (square function x2, cube function x3)
differentiate functions with coefficients dxd(cf(x))=cdxdf(x), c is a constant (2x, 3x)
differentiate combinations of functions dxd(f(x)±g(x))=dxdf(x)±dxdg(x) (polynomials, trigonometric functions)
more complex functions dxd(f(x)g(x))=f(x)dxdg(x)+g(x)dxdf(x) (polynomial multiplied by exponential function)
more complex functions dxd(g(x)f(x))=[g(x)]2g(x)dxdf(x)−f(x)dxdg(x) (rational functions)
Chain rule composite functions dxdf(g(x))=f′(g(x))g′(x) (function inside another function like sin(x2))
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)=dtds(t)
Find acceleration by taking second derivative of position or first derivative of velocity acceleration function: a(t)=dt2d2s(t)=dtdv(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
Identify given information and desired quantity to solve for (initial position, velocity, time)
Set up position function based on problem description (quadratic for constant acceleration, trigonometric for periodic motion)
Use differentiation rules to find velocity and acceleration functions
Evaluate functions at specific times or solve for unknown variables (position at t=5 seconds, time when velocity is zero)
Interpret results in context of problem
Positive velocity indicates motion in positive direction, negative velocity indicates motion in negative direction