Calculus Derivatives Rules

Derivadas

Reglas de Derivación

1. Regla de la Potencia

   • ( f(x) = x^n )
   • ( f'(x) = n \cdot x^{n-1} )

2. Regla de la Suma

   • Si ( f(x) = g(x) + h(x) )
   • ( f'(x) = g'(x) + h'(x) )

3. Regla de la Resta

   • Si ( f(x) = g(x) - h(x) )
   • ( f'(x) = g'(x) - h'(x) )

4. Regla del Producto

   • Si ( f(x) = g(x) \cdot h(x) )
   • ( f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) )

5. Regla del Cociente

   • Si ( f(x) = \frac{g(x)}{h(x)} )
   • ( f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2} )

6. Regla de la Cadena

   • Si ( y = f(u) ) y ( u = g(x) )
   • ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = f'(g(x)) \cdot g'(x) )

7. Derivadas de Funciones Trigonométricas

   • ( \frac{d}{dx}(\sin x) = \cos x )
   • ( \frac{d}{dx}(\cos x) = -\sin x )
   • ( \frac{d}{dx}(\tan x) = \sec^2 x )

8. Derivadas de Funciones Exponenciales y Logarítmicas

   • ( \frac{d}{dx}(e^x) = e^x )
   • ( \frac{d}{dx}(a^x) = a^x \ln(a) )
   • ( \frac{d}{dx}(\ln x) = \frac{1}{x} )

9. Derivadas de Funciones Inversas

   • Si ( y = f^{-1}(x) ), entonces ( \frac{dy}{dx} = \frac{1}{f'(y)} )

Notas Adicionales

• Es importante recordar las condiciones de continuidad y diferenciabilidad.
• La regla de la cadena es especialmente útil para funciones compuestas.
• Practicar la derivación utilizando estas reglas facilitará la comprensión de conceptos más avanzados.

Derivatives

Differentiation Rules

• Power Rule: If ( f(x) = x^n ), then the derivative is ( f'(x) = n \cdot x^{n-1} ).
• Sum Rule: For functions ( f(x) = g(x) + h(x) ), the derivative is ( f'(x) = g'(x) + h'(x) ).
• Difference Rule: For functions ( f(x) = g(x) - h(x) ), the derivative is ( f'(x) = g'(x) - h'(x) ).
• Product Rule: If ( f(x) = g(x) \cdot h(x) ), then ( f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) ).
• Quotient Rule: For ( f(x) = \frac{g(x)}{h(x)} ), the derivative is ( f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2} ).
• Chain Rule: For ( y = f(u) ) and ( u = g(x) ), the derivative is ( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) ).

Trigonometric Functions

• Derivative of ( \sin x ) is ( \cos x ).
• Derivative of ( \cos x ) is ( -\sin x ).
• Derivative of ( \tan x ) is ( \sec^2 x ).

Exponential and Logarithmic Functions

• Derivative of ( e^x ) is ( e^x ).
• Derivative of ( a^x ) is ( a^x \ln(a) ).
• Derivative of ( \ln x ) is ( \frac{1}{x} ).

Inverse Functions

• For ( y = f^{-1}(x) ), the derivative is ( \frac{dy}{dx} = \frac{1}{f'(y)} ).

Additional Notes

• Continuity and differentiability are essential conditions for applying these rules.
• The chain rule is particularly useful for composite functions.
• Practicing differentiation using these rules enhances understanding of advanced concepts.

Derivative Rules

• Power Rule: For ( f(x) = x^n ), the derivative is ( f'(x) = n \cdot x^{n-1} ).

• Sum Rule: For ( f(x) = g(x) + h(x) ), the derivative is ( f'(x) = g'(x) + h'(x) ).

• Difference Rule: For ( f(x) = g(x) - h(x) ), the derivative is ( f'(x) = g'(x) - h'(x) ).

• Product Rule: For ( f(x) = g(x) \cdot h(x) ), the derivative is ( f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) ).

• Quotient Rule: For ( f(x) = \frac{g(x)}{h(x)} ), the derivative is ( f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2} ).

• Chain Rule: For a composite function ( f(g(x)) ), the derivative is ( f'(g(x)) \cdot g'(x) ).

Trigonometric Derivatives

• Derivative of sine: ( \frac{d}{dx}(\sin x) = \cos x )
• Derivative of cosine: ( \frac{d}{dx}(\cos x) = -\sin x )
• Derivative of tangent: ( \frac{d}{dx}(\tan x) = \sec^2 x )

Exponential and Logarithmic Derivatives

• Derivative of an exponential function: ( \frac{d}{dx}(a^x) = a^x \cdot \ln(a) )
• Derivative of the natural logarithm: ( \frac{d}{dx}(\ln x) = \frac{1}{x} )

Inverse Function Derivative

• For ( y = f^{-1}(x) ), the derivative is ( \frac{dy}{dx} = \frac{1}{f'(y)} ).

Composition of Functions

• Apply the chain rule alongside other rules as required for composed functions.

Additional Notes

• Differentiation relates to limits; understanding this context is crucial.
• Familiarize with notation such as ( f'(x) ) and ( \frac{dy}{dx} ).
• Practice the application of these rules across various function combinations to gain proficiency.