Common Derivative Rules

Understanding common derivative rules is key in Calculus I. These rules simplify the process of finding derivatives, making it easier to analyze functions. Mastering them will help you tackle more complex problems in analytic geometry and calculus with confidence.

1. Constant Rule

   • The derivative of a constant function is zero.
   • This rule applies to any constant value, such as 5, -3, or π.
   • It simplifies the differentiation process by indicating that constants do not change.

2. Power Rule

   • The derivative of ( x^n ) is ( n \cdot x^{n-1} ), where ( n ) is any real number.
   • This rule is essential for differentiating polynomial functions.
   • It allows for quick calculations of derivatives for powers of ( x ).

3. Sum and Difference Rule

   • The derivative of a sum (or difference) of functions is the sum (or difference) of their derivatives.
   • If ( f(x) = g(x) + h(x) ), then ( f'(x) = g'(x) + h'(x) ).
   • This rule facilitates the differentiation of complex functions by breaking them down into simpler parts.

4. Product Rule

   • The derivative of a product of two functions is given by ( (u \cdot v)' = u'v + uv' ).
   • This rule is crucial when dealing with products of functions, ensuring both functions are accounted for.
   • It emphasizes the need to differentiate each function while keeping the other intact.

5. Quotient Rule

   • The derivative of a quotient of two functions is given by ( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} ).
   • This rule is essential for differentiating functions that are expressed as fractions.
   • It highlights the importance of both the numerator and denominator in the differentiation process.

6. Chain Rule

   • The derivative of a composite function is given by ( (f(g(x)))' = f'(g(x)) \cdot g'(x) ).
   • This rule is vital for functions that are nested within one another.
   • It allows for the differentiation of complex functions by treating the inner function separately.

7. Exponential Function Rule

   • The derivative of ( e^x ) is ( e^x ), and for ( a^x ) (where ( a ) is a constant), it is ( a^x \ln(a) ).
   • This rule is fundamental for working with exponential growth and decay models.
   • It simplifies the differentiation of exponential functions, which are prevalent in calculus.

8. Natural Logarithm Rule

   • The derivative of ( \ln(x) ) is ( \frac{1}{x} ) for ( x > 0 ).
   • This rule is essential for functions involving natural logarithms.
   • It provides a straightforward method for differentiating logarithmic functions.

9. Trigonometric Function Rules (sin, cos, tan)

   • The derivatives are: ( \frac{d}{dx}(\sin(x)) = \cos(x) ), ( \frac{d}{dx}(\cos(x)) = -\sin(x) ), and ( \frac{d}{dx}(\tan(x)) = \sec^2(x) ).
   • These rules are crucial for analyzing periodic functions and their rates of change.
   • They form the foundation for calculus applications in physics and engineering.

10. Inverse Trigonometric Function Rules

    • The derivatives are: ( \frac{d}{dx}(\arcsin(x)) = \frac{1}{\sqrt{1-x^2}} ), ( \frac{d}{dx}(\arccos(x)) = -\frac{1}{\sqrt{1-x^2}} ), and ( \frac{d}{dx}(\arctan(x)) = \frac{1}{1+x^2} ).
    • These rules are essential for differentiating inverse trigonometric functions.
    • They are particularly useful in integration and solving equations involving angles.