BackEssential Derivative and Integration Rules with Applications
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Derivative Rules and Their Applications
Basic Derivative Rules
The derivative of a function measures the rate at which the function changes with respect to its variable. Several fundamental rules allow us to compute derivatives efficiently.
Power Rule: For , the derivative is .
Sum Rule: For , the derivative is .
Product Rule: For , the derivative is .
Quotient Rule: For , the derivative is .
Chain Rule: For , the derivative is .
Derivatives of Common Functions
Natural Exponential: , .
Natural Logarithm: , .
Sine: , .
Cosine: , .
Tangent: , .
Secant: , .
Inverse Sine: , .
Inverse Tangent: , .
Examples of Derivative Computation
Example 1:
Example 2:
Example 3:
Example 4:
Higher Order Derivatives
Example: First derivative: Second derivative:
Applications of Derivatives
Tangent Line Approximation
The tangent line to a function at a point provides a linear approximation near that point.
Equation of Tangent Line: For at , .
Example: at , Tangent line: Approximate :
Integration and Antiderivatives
Relationship Between Derivatives and Integrals
If is the derivative of , then is an antiderivative of . The indefinite integral of is .
if and only if
Indefinite Integrals: Examples
Example (a):
Example (b): Expand numerator:
Example (c):
Example (d):
Table: Derivative Rules Summary
Rule Category | Function | Derivative |
|---|---|---|
Power Rule | ||
Sum Rule | ||
Product Rule | ||
Quotient Rule | ||
Chain Rule | ||
Natural Exponential | ||
Natural Logarithm | ||
Sine | ||
Cosine | ||
Tangent | ||
Secant | ||
Inverse Sine | ||
Inverse Tangent |
Additional info:
Some expressions were clarified and expanded for academic completeness.
Where original notation was ambiguous, standard calculus notation was used.
Examples were rewritten for clarity and accuracy.
