Math Problem Statement
cari derivatif pertama
cari derivatif pertama
Solution
The given function is:
First, let's simplify the expression:
Now, since , we need to take the derivative considering the absolute value function.
The derivative of is given by:
\begin{cases} 2, & \text{if } x > 0 \\ -2, & \text{if } x < 0 \\ \text{undefined}, & \text{if } x = 0 \end{cases}$$ Therefore, the first derivative of the function is piecewise: $$f'(x) = \begin{cases} 2, & \text{if } x > 0 \\ -2, & \text{if } x < 0 \end{cases}$$ And at $$x = 0$$, the derivative is undefined because of the non-differentiability of the absolute value function at that point. Would you like more details or have any questions? Here are five follow-up questions: 1. How do we handle derivatives of absolute value functions in general? 2. Why is the derivative undefined at $$x = 0$$? 3. How can we find the second derivative of $$f(x)$$? 4. What happens if $$f(x)$$ was defined differently for negative $$x$$? 5. How does the derivative change if the function was $$f(x) = \sqrt{4x^4}$$ instead? **Tip:** When taking derivatives involving absolute values, always consider the piecewise definition of the function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivative
Absolute Value Function
Piecewise Function
Formulas
f'(x) = 2 for x > 0
f'(x) = -2 for x < 0
Theorems
Differentiability of absolute value functions
Piecewise differentiation
Suitable Grade Level
Grades 11-12 (High School Calculus)