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.

Test your understanding of derivative rules including the power, sum, subtraction, product, and quotient rules. This quiz will assess your ability to apply these fundamental principles of calculus effectively. Prepare to tackle problems that challenge your grasp of derivatives!