Physics
Calculus
Differentiation
11
⚡ Quick Summary
Differentiation is finding the rate of change of a function. It tells you how much the output changes for a tiny change in the input.
Product Rule: d/dx (u*v) = u(dv/dx) + v(du/dx)
Quotient Rule: d/dx (u/v) = [v(du/dx) - u(dv/dx)] / v²
Chain Rule: d/dx f(g(x)) = f'(g(x)) * g'(x)
The text provides examples of finding derivatives of functions like x²sin(x), sin(x)/x, and sin(x²). It uses the product rule, quotient rule, and chain rule of differentiation.
Finding Maximum/Minimum Values
11
⚡ Quick Summary
To find where a function is at its highest or lowest, find where its derivative equals zero. Also, check the endpoints and consider the behavior of the function at extreme values of x.
If dy/dx = 0, then y is at a maximum or minimum.
The text demonstrates how to find the maximum or minimum value of a function by setting its derivative equal to zero and solving for x. It also considers the behavior of the function as x approaches extreme values to determine whether the critical point corresponds to a maximum or minimum.