Question
upstudy study bank question image url

Differentiate. \[ f(t)=\frac{2 t}{9+t^{2}} \] \( f^{\prime}(t)=\square \)

Ask by Mccoy Love. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\( f^{\prime}(t)=\frac{2(9-t^2)}{(9+t^2)^2} \)

Solution

1. Recall the quotient rule for differentiation: if \[ f(t)=\frac{g(t)}{h(t)}, \] then \[ f'(t)=\frac{g'(t)h(t)-g(t)h'(t)}{[h(t)]^2}. \] 2. Identify \( g(t) \) and \( h(t) \): \[ g(t)=2t \quad \text{and} \quad h(t)=9+t^2. \] 3. Differentiate \( g(t) \) and \( h(t) \): \[ g'(t)=2, \] \[ h'(t)=2t. \] 4. Substitute into the quotient rule: \[ f'(t)=\frac{2(9+t^2)-2t(2t)}{(9+t^2)^2}. \] 5. Simplify the numerator: \[ 2(9+t^2)=18+2t^2, \] and \[ 2t(2t)=4t^2. \] Thus, \[ 18+2t^2-4t^2 = 18-2t^2. \] 6. Factor the common factor in the numerator: \[ 18-2t^2=2(9-t^2). \] 7. The derivative is: \[ f'(t)=\frac{2(9-t^2)}{(9+t^2)^2}. \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Mind Expander

To differentiate the function \( f(t) = \frac{2t}{9+t^2} \), we can use the quotient rule, which states that if \( f(t) = \frac{g(t)}{h(t)} \), then \( f'(t) = \frac{g'(t)h(t) - g(t)h'(t)}{(h(t))^2} \). Here, \( g(t) = 2t \) and \( h(t) = 9+t^2 \). Calculating the derivatives: - \( g'(t) = 2 \) - \( h'(t) = 2t \) Now, apply the quotient rule: \[ f'(t) = \frac{(2)(9+t^2) - (2t)(2t)}{(9+t^2)^2} = \frac{(18 + 2t^2 - 4t^2)}{(9+t^2)^2} = \frac{18 - 2t^2}{(9+t^2)^2} \] So, the final answer is: \[ f'(t) = \frac{18 - 2t^2}{(9+t^2)^2} \]

Related Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy