If f(t) = 5 ln (t) - 3e-2t , what is f '(t) ?

To find \( f'(t) \), we need to differentiate the given function:

\[
f(t) = 5 \ln(t) - 3e^{-2t}
\]

### Step-by-step differentiation:

1. **Differentiate \( 5 \ln(t) \):**

   The derivative of \( \ln(t) \) is \( \frac{1}{t} \). So:
   \[
   \frac{d}{dt} \left( 5 \ln(t) \right) = \frac{5}{t}
   \]

2. **Differentiate \( -3e^{-2t} \):**

   The derivative of \( e^{u(t)} \) where \( u(t) \) is a function of \( t \), is \( e^{u(t)} \cdot u'(t) \). Here, \( u(t) = -2t \), so:
   \[
   \frac{d}{dt} \left( -3e^{-2t} \right) = -3 \cdot e^{-2t} \cdot (-2) = 6e^{-2t}
   \]

### Final result:

Now, putting the two derivatives together:

\[
f'(t) = \frac{5}{t} + 6e^{-2t}
\]

Would you like more details or have any questions?

Here are 5 related questions for deeper exploration:

1. How do you find the second derivative \( f''(t) \) of this function?
2. How does the natural logarithm \( \ln(t) \) behave as \( t \to 0^+ \)?
3. What is the behavior of \( f(t) \) as \( t \to \infty \)?
4. How do you find the critical points of the function \( f(t) \)?
5. What does the graph of \( f'(t) \) look like?

**Tip:** When differentiating exponential functions, always pay attention to the chain rule if the exponent involves a variable.

If f(t) = 5 ln (t) - 3e-2t, what is f '(t)?

Learn how to differentiate the function f(t) = 5 ln(t) - 3e^-2t step-by-step. This calculus problem explores key concepts such as the chain rule, logarithmic differentiation, and exponential functions. Suitable for Grades 11-12 students studying advanced calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation