Math Problem Statement
Derivative of f(x)=x|x| at 1
Solution
To find the derivative of the function at , we need to break the function into cases due to the absolute value.
Step 1: Rewrite the function using piecewise notation
The function can be written as:
\begin{cases} x^2, & \text{if } x \geq 0 \\ -x^2, & \text{if } x < 0 \end{cases}$$ This piecewise definition makes it easier to handle the derivative. ### Step 2: Differentiate each piece Now, we differentiate each piece of the function separately: - For $$x \geq 0$$, $$f(x) = x^2$$, so: $$f'(x) = 2x$$ - For $$x < 0$$, $$f(x) = -x^2$$, so: $$f'(x) = -2x$$ ### Step 3: Evaluate the derivative at $$x = 1$$ Since $$1 \geq 0$$, we use the derivative from the $$x \geq 0$$ case, which is $$f'(x) = 2x$$. Therefore, at $$x = 1$$: $$f'(1) = 2(1) = 2$$ Thus, the derivative of $$f(x) = x|x|$$ at $$x = 1$$ is $$f'(1) = 2$$. --- Would you like more details on the differentiation process, or do you have any questions? Here are 5 related questions you might find interesting: 1. What is the derivative of $$f(x) = x|x|$$ at $$x = -1$$? 2. How would you compute the second derivative of $$f(x) = x|x|$$? 3. Can we use the chain rule to differentiate absolute value functions? 4. What is the general formula for the derivative of $$|x|$$? 5. How do you differentiate functions with absolute values at points of non-differentiability? **Tip**: Always check the continuity and differentiability of piecewise functions at the boundary points, such as where the absolute value changes definition!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Differentiation
Piecewise Functions
Absolute Value
Formulas
f'(x) = 2x for x ≥ 0
f'(x) = -2x for x < 0
Theorems
Piecewise Differentiation
Basic Derivative Rules
Suitable Grade Level
Grades 11-12 or Early College