BackEssential Derivative Rules and Related Formulas in Calculus I
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Derivatives
Basic Derivative Rules
Derivatives measure the rate at which a function changes as its input changes. The following are fundamental rules for differentiating common functions:
Power Rule: For any real number r, the derivative of is given by:
Sine Function:
Cosine Function:
Tangent Function:
Cotangent Function:
Secant Function:
Cosecant Function:
Product Rule
The product rule is used to differentiate the product of two functions:
Formula:
Example: If and , then
Quotient Rule
The quotient rule is used to differentiate the quotient of two functions:
Formula:
Example: If and , then
Chain Rule
The chain rule is used to differentiate composite functions:
Formula:
Common Use: For ,
Example: If , then
Linearization
Linear Approximation of a Function
Linearization provides an approximation of a function near a given point using the tangent line at that point.
Formula:
Example: To approximate near : Let , , , so . Linearization:
Quadratic Formula
Solving Quadratic Equations
The quadratic formula provides the solutions to any quadratic equation of the form .
Formula:
Example: Solve : , ,
