Product and Quotient Rules
Product and Quotient Rules
Differentiation Basics
- Differentiation is the process of finding the derivative of a function, representing the rate at which the function is changing.
- Constant functions differentiate to zero, since there is no change.
- When differentiating linear functions, the derivative is the coefficient of x.
Power Rule
- The power rule states that if you have a function in the form x^n, the derivative is n*x^(n-1).
- Remember to reduce the power by 1 and multiply by the original power.
Stationary Points
- Stationary points are points where the derivative is zero, indicating a local maximum, minimum or more rarely a point of inflexion.
- To classify a stationary point, we can use the second derivative test. If the second derivative is positive, the stationary point is a minimum. If the second derivative is negative, the stationary point is a maximum.
Convex and Concave Curves
- The second derivative can tell us whether a function is convex or concave.
- If the second derivative is positive, the function is convex (also called concave up). If the second derivative is negative, the function is concave (also called concave down).
Using Differentiation
- Differentiation can be used to solve many kinds of problems including finding the slope of a curve, finding local maxima and minima, and finding points of inflexion.
- Real-world applications of differentiation include physical science, engineering, economics, statistics, and many other disciplines.
Chain Rule
- The chain rule is used when differentiating a function that is made up of two or more functions.
- In essence, the derivative of the outside function is multiplied by the derivative of the inside function.
Differentiating ex, ln x, and axe
- The derivative of e^x is e^x.
- The derivative of ln(x) is 1/x.
- The derivative of a^x is a^x * ln(a).
Differentiating sin, cos and tan
- The derivative of sin(x) is cos(x).
- The derivative of cos(x) is -sin(x).
- The derivative of tan(x) is sec^2(x).
Product and Quotient Rules
- The product rule is used when differentiating the product of two functions.
- The quotient rule is used when differentiating the quotient of two functions.