Math Problem Statement

What is the derivative of g(x) = arcsin(e^(4x)) + 3?

Solution

We are given the function g(x)=arcsin(e4x)+3g(x) = \arcsin(e^{4x}) + 3, and the task is to find the derivative of g(x)g(x).

Step-by-Step Solution:

  1. The derivative of the constant 33 is 00, so we can focus on differentiating arcsin(e4x)\arcsin(e^{4x}).

  2. The derivative of arcsin(u)\arcsin(u) with respect to xx is given by: ddx[arcsin(u)]=11u2dudx\frac{d}{dx} [\arcsin(u)] = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} where u=e4xu = e^{4x} in this case.

  3. Now, calculate the derivative of u=e4xu = e^{4x}: dudx=4e4x\frac{du}{dx} = 4e^{4x}

  4. Applying the chain rule, we get: ddxarcsin(e4x)=11(e4x)24e4x\frac{d}{dx} \arcsin(e^{4x}) = \frac{1}{\sqrt{1 - (e^{4x})^2}} \cdot 4e^{4x}

  5. Simplify the expression inside the square root: (e4x)2=e8x(e^{4x})^2 = e^{8x} So, the derivative becomes: ddxg(x)=4e4x1e8x\frac{d}{dx} g(x) = \frac{4e^{4x}}{\sqrt{1 - e^{8x}}}

Thus, the derivative of g(x)g(x) is: g(x)=4e4x1e8xg'(x) = \frac{4e^{4x}}{\sqrt{1 - e^{8x}}}

Let me know if you need any further details!


Here are five related questions for you to consider:

  1. How does the derivative of the arcsin\arcsin function change if the argument involves different powers of exe^x?
  2. Can you compute the second derivative of the function g(x)g(x)?
  3. What is the derivative of g(x)=arcsin(e4x)+3g(x) = \arcsin(e^{4x}) + 3 evaluated at x=0x = 0?
  4. How would you find the critical points of the function g(x)g(x)?
  5. How does the behavior of the function g(x)g(x) change as xx \to \infty?

Tip: When differentiating composite functions like arcsin(f(x))\arcsin(f(x)), always remember to use the chain rule effectively!

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Chain Rule
Inverse Trigonometric Functions

Formulas

d/dx [arcsin(u)] = 1 / sqrt(1 - u^2) * du/dx
d/dx [e^(4x)] = 4e^(4x)

Theorems

Chain Rule
Derivative of Inverse Trigonometric Functions

Suitable Grade Level

University/Advanced High School (Grades 11-12)