Math Problem Statement

Find the derivative of the function F(t) = e^{6t \sin(2t)}.

Solution

To find the derivative of the function F(t)=e6tsin(2t)F(t) = e^{6t \sin(2t)}, we need to use the chain rule. Here’s the step-by-step solution:

Step-by-Step Solution

  1. Identify the outer and inner functions:

    • The outer function is eue^u where u=6tsin(2t)u = 6t \sin(2t).
    • The inner function is u=6tsin(2t)u = 6t \sin(2t).
  2. Differentiate the outer function eue^u with respect to uu: ddu(eu)=eu\frac{d}{du} (e^u) = e^u

  3. Differentiate the inner function 6tsin(2t)6t \sin(2t) with respect to tt using the product rule: u=ddt(6tsin(2t))=6sin(2t)+6tddt(sin(2t))u' = \frac{d}{dt} (6t \sin(2t)) = 6 \sin(2t) + 6t \cdot \frac{d}{dt} (\sin(2t))

    • Here, ddt(sin(2t))=2cos(2t)\frac{d}{dt} (\sin(2t)) = 2 \cos(2t) (using the chain rule again). u=6sin(2t)+6t2cos(2t)=6sin(2t)+12tcos(2t)u' = 6 \sin(2t) + 6t \cdot 2 \cos(2t) = 6 \sin(2t) + 12t \cos(2t)
  4. Combine using the chain rule:

    • Now we apply the chain rule: F(t)=e6tsin(2t)uF'(t) = e^{6t \sin(2t)} \cdot u'. F(t)=e6tsin(2t)(6sin(2t)+12tcos(2t))F'(t) = e^{6t \sin(2t)} \cdot (6 \sin(2t) + 12t \cos(2t))

Thus, the derivative of the function is: F(t)=e6tsin(2t)(6sin(2t)+12tcos(2t))F'(t) = e^{6t \sin(2t)} \left(6 \sin(2t) + 12t \cos(2t)\right)

Would you like more details on this solution or have any questions?

Related Questions:

  1. How do you apply the product rule for derivatives?
  2. What are the steps to apply the chain rule to composite functions?
  3. How does the derivative of sin(kt)\sin(kt) differ from that of sin(t)\sin(t)?
  4. What is the significance of using the chain rule in differentiation?
  5. Can you show an example of differentiating a function with a nested exponent?

Tip:

When differentiating exponential functions with a complex exponent, breaking it down using the chain rule makes it much easier to handle.

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Exponential Functions
Chain Rule
Product Rule

Formulas

Chain Rule: (f(g(t)))' = f'(g(t)) * g'(t)
Product Rule: (uv)' = u'v + uv'

Theorems

Chain Rule
Product Rule

Suitable Grade Level

Grades 11-12 or early university