Pregunta
upstudy study bank question image url

Video Example4)
Find an equation of the tangent line to the curve at the point ,
Solution
According to the Quotient Rule, we have
So the slope of the tangent line at is
This means that the tangeht line at is horizontal and its equation is

Ask by George Delgado. in the United States
Mar 15,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The equation of the tangent line at the point is .

Solución

We are given the function
and we need to find the equation of the tangent line at the point .

Step 1. Compute the derivative

Let
Then, their derivatives are
Using the Quotient Rule,
we substitute , , , and :

Step 2. Simplify the derivative

Factor from the numerator:
The expression simplifies to
Thus, the derivative is

Step 3. Evaluate the derivative at

Substitute into the derivative:
Compute the expression inside the parentheses:
So the numerator becomes
And the denominator is
Thus,

Step 4. Write the equation of the tangent line

The slope of the tangent line at is ; hence, the tangent line is horizontal. Its equation is therefore

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Extra Insights

Picture this: the Quotient Rule in calculus is like following a recipe that combines two delicious ingredients! When you’re differentiating a function like , the key is to remember that you have to apply both the derivative of the top and the bottom. Get your brain ready to unleash some derivative magic while keeping that fraction in check!
Now, let’s talk about what happens when you do this process correctly. Once you evaluate the derivative at the point , you can find the slope of that tangent line. If your calculation shows that the slope is zero, congrats! You just discovered that your tangent line is horizontal, and you’ll get to write the magical equation . That’s like drawing a straight line in the sky to keep your path clear for all your mathematical adventures!

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad