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Calculus Learning Center
Differentiation Rules
The power rule is used to find the derivative of any function in the form x raised to a constant power. Multiply by the power, then reduce the power by 1.
The product rule is used when differentiating the product of two functions. The derivative is the first function’s derivative times the second function, plus the first function times the second function’s derivative.
The quotient rule is used for differentiating the quotient of two functions. The derivative is the denominator times the numerator’s derivative, minus the numerator times the denominator’s derivative, all divided by the square of the denominator.
The chain rule is used when differentiating a composite function. The derivative is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
These are the standard derivatives for the six basic trigonometric functions. Remember to apply the chain rule when the argument is not simply x.
Applications of Derivatives
Finding Rate of Change
Derivatives represent the rate at which a function changes. This is useful in physics (velocity and acceleration), economics (marginal cost and revenue), and biology (population growth rates).
Optimization Problems
By finding where a derivative equals zero, we can determine the maximum or minimum values of a function. This helps solve problems like maximizing profit or minimizing cost.
Approximating Functions
Through Taylor and Maclaurin series, derivatives help approximate complex functions using polynomial expressions, making calculations easier.
Analyzing Curve Behavior
The first derivative tells us where a function is increasing or decreasing, while the second derivative reveals information about concavity and inflection points.
Common Derivative Functions
| Function | Derivative |
|---|---|
| c (constant) | 0 |
| xn | nxn-1 |
| ex | ex |
| ln(x) | 1/x |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec2(x) |
| ax | ax·ln(a) |
| loga(x) | 1/(x·ln(a)) |
Derivatives help us understand how quantities change with respect to each other. They’re essential in physics to describe motion, in economics to analyze marginality, in engineering for optimization, and in many other fields where rates of change are important.
Explicit differentiation is used when y is explicitly expressed as a function of x (like y = x²). Implicit differentiation is used when the relationship between x and y is given implicitly (like x² + y² = 1). In implicit differentiation, we treat y as a function of x and use the chain rule when differentiating terms with y.
The second derivative is the derivative of the first derivative. It measures how the rate of change is itself changing—essentially, the acceleration of the function. It’s used to determine concavity, inflection points, and in applications like understanding acceleration in physics.
Critical points occur where the derivative equals zero or is undefined. They are candidates for extrema (maximum or minimum values). To determine if a critical point is a maximum, minimum, or neither, we use the second derivative test or the first derivative test. If the second derivative is negative at a critical point, it’s a maximum; if positive, it’s a minimum.