Calculus Derivatives Rules

Questions and Answers

¿Cuál es la derivada de la función $f(x) = x^3$ utilizando la regla de potencia?

$3x^3$

$4x^3$

$2x^3$

$3x^2$ (correct)

Si $f(x) = g(x) + h(x)$, ¿cómo se expresa la derivada $f'(x)$ según la regla de suma?

$g(x) + h'(x)$

$g'(x) + h'(x)$ (correct)

$g'(x) + h(x)$

$g(x) - h'(x)$

¿Qué establece la regla del cociente para la derivada de $f(x) = \frac{g(x)}{h(x)}$?

$\frac{g'(x) imes h(x) - g(x) imes h'(x)}{(h(x))^2}$ (correct)

$\frac{g'(x) imes h'(x)}{h(x)}$

$\frac{g'(x) imes h(x)}{(h(x))^2}$

$g'(x) - h'(x)$

Al derivar la función $f(x) = an x$, ¿cuál es el resultado correcto?

$\sec^2 x$ Signup and view all the answers

¿Cuál es la derivada de $f(x) = a^x$ según las derivadas de funciones exponenciales?

$a^x \cdot \ln(a)$ Signup and view all the answers

Utilizando la regla de la cadena, ¿cómo se presenta la derivada de la función compuesta $f(g(x))$?

$f'(g(x)) \cdot g'(x)$ Signup and view all the answers

Study Notes

Derivadas

Reglas de Derivación

Regla de la Potencia
- ( f(x) = x^n )
- ( f'(x) = n \cdot x^{n-1} )
Regla de la Suma
- Si ( f(x) = g(x) + h(x) )
- ( f'(x) = g'(x) + h'(x) )
Regla de la Resta
- Si ( f(x) = g(x) - h(x) )
- ( f'(x) = g'(x) - h'(x) )
Regla del Producto
- Si ( f(x) = g(x) \cdot h(x) )
- ( f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) )
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} )
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) )
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 )
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} )
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.

Description

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!