derivative of x to the x
Understand the Problem
The question is asking for the derivative of the function y = x^x with respect to x. To solve this, we will apply logarithmic differentiation.
Answer
$\frac{dy}{dx} = x^x \cdot (1 + \text{ln}(x))$
Answer for screen readers
The final answer is $\frac{dy}{dx} = x^x \cdot (1 + \text{ln}(x))$
Steps to Solve
- Take the natural logarithm of both sides
To simplify the differentiation, take the natural logarithm of both sides of the equation $y = x^x$.
$$ ext{Let } y = x^x$$
$$ ext{Then, } \ ext{ln}(y) = ext{ln}(x^x)$$
- Use the property of logarithms
Use the logarithm property $\text{ln}(a^b) = b \cdot \text{ln}(a)$ to simplify the right-hand side.
$$ ext{ln}(y) = x \cdot \text{ln}(x)$$
- Differentiate implicitly with respect to x
Differentiate both sides with respect to $x$.
$$\frac{d}{dx} \left( ext{ln}(y) \right) = \frac{d}{dx} \left( x \cdot \text{ln}(x) \right)$$
Using the chain rule on the left side:
$$\frac{1}{y} \cdot \frac{dy}{dx} = x \cdot \frac{1}{x} + \text{ln}(x) \cdot \frac{d}{dx}(x)$$
- Simplify the right-hand side
Simplify the terms on the right-hand side of the equation.
$$\frac{1}{y} \cdot \frac{dy}{dx} = 1 + \text{ln}(x)$$
- Isolate the derivative $\frac{dy}{dx}$
Finally, solve for $\frac{dy}{dx}$ by multiplying both sides by $y$.
$$\frac{dy}{dx} = y \cdot (1 + \text{ln}(x))$$
Since $y = x^x$:
$$\frac{dy}{dx} = x^x \cdot (1 + \text{ln}(x))$$
The final answer is $\frac{dy}{dx} = x^x \cdot (1 + \text{ln}(x))$
More Information
Logarithmic differentiation is a useful tool for finding the derivatives of functions where both the base and the exponent are variables.
Tips
A common mistake is forgetting to multiply by $y$ after differentiating implicitly. Always remember that $y = x^x$, and substitute back the original function at the end.
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