Using Differentiation

Differentiation is a process that takes a function and produces its derivative.
The derivative measures the rate at which the original function changes.
For a function f(x), its derivative is represented as f’(x) or df/dx.

Power Rule: The derivative of x^n, where n is any real number, is n*x^(n-1).
Product Rule: The derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.
Chain Rule: Used for composing derivatives of functions inside other functions. Simplified, it’s the derivative of the outer function, multiplied by the derivative of the inner function.
Quotient Rule: If a function is the ratio of two other functions, its derivative can be found using this rule.

Stationary points are points where the derivative of a function equals zero.
They come in three types: local minimum, local maximum, and inflexion point.
To classify a stationary point, you use the second derivative test.

A curve is convex if its second derivative is positive, and concave if its second derivative is negative.
At a point of inflexion, the curve changes from being concave to convex, or vice versa.

Optimization: Use differentiation to find maximum or minimum values of a function, important for finding most efficient solutions in engineering, economics, etc.
Related Rates: With differentiation, you can relate the rates at which quantities change; important in physics for understanding velocities and accelerations.
Graphing Functions: Differentiation helps identify key features of a function’s graph, like where it reaches its highest or lowest points.