Given that y = x ln(x), find dy/dx.
Understand the Problem
The question is asking to differentiate the function y = x ln(x) with respect to x. To find dy/dx, we will apply the product rule of differentiation since the function is a product of two functions: x and ln(x).
Answer
The derivative is $$ \frac{dy}{dx} = \ln(x) + 1 $$
Answer for screen readers
The derivative of the function $y = x \ln(x)$ with respect to $x$ is:
$$ \frac{dy}{dx} = \ln(x) + 1 $$
Steps to Solve
-
Identify the functions to differentiate
The given function is $y = x \ln(x)$. We will consider two functions:- $u = x$
- $v = \ln(x)$
-
Apply the product rule
The product rule states that if you have two functions $u(x)$ and $v(x)$, the derivative of their product is given by:
$$ \frac{dy}{dx} = \frac{du}{dx} v + u \frac{dv}{dx} $$ -
Differentiate the individual functions
Now, we will compute the derivatives of $u$ and $v$:- For $u = x$, the derivative $\frac{du}{dx} = 1$.
- For $v = \ln(x)$, the derivative $\frac{dv}{dx} = \frac{1}{x}$.
-
Substitute into the product rule formula
Now substitute the derivatives and functions into the product rule:
$$ \frac{dy}{dx} = 1 \cdot \ln(x) + x \cdot \frac{1}{x} $$ -
Simplify the expression
Next, simplify the expression:
$$ \frac{dy}{dx} = \ln(x) + 1 $$
The derivative of the function $y = x \ln(x)$ with respect to $x$ is:
$$ \frac{dy}{dx} = \ln(x) + 1 $$
More Information
The derivative $\frac{dy}{dx} = \ln(x) + 1$ indicates how the function $y = x \ln(x)$ changes with respect to $x$. This function combines a linear term with a logarithmic term, making it useful in various applications such as economics and data analysis.
Tips
- Forgetting to apply the product rule correctly. Make sure to identify both functions in the product.
- Not differentiating the logarithmic function correctly. Remember that the derivative of $\ln(x)$ is $\frac{1}{x}$.
AI-generated content may contain errors. Please verify critical information
