Find dy/dx y=x natural log of x-x

Solution:

Step:1

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y)=\frac{d}{dx}(x\ln (x)-x)$.

Step:2

The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.

Step:3

Proceed to differentiate the expression on the right-hand side.

Step:3.1

Apply the Sum Rule to find the derivative of the sum and difference: $\frac{d}{dx}(x\ln (x))+\frac{d}{dx}(-x)$.

Step:3.2

Compute $\frac{d}{dx}(x\ln (x))$.

Step:3.2.1

Use the Product Rule, which is given by $\frac{d}{dx}(uv)=u\frac{d}{dx}(v)+v\frac{d}{dx}(u)$, where $u=x$ and $v=\ln (x)$.

Step:3.2.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$, thus we have $x\cdot \frac{1}{x}+\ln (x)\cdot \frac{d}{dx}(x)+\frac{d}{dx}(-x)$.

Step:3.2.3

Apply the Power Rule, which states that $\frac{d}{dx}(x^n)=n{x}^{n-1}$, where $n=1$.

Step:3.2.4

Simplify the expression by combining $x$ and $\frac{1}{x}$.

Step:3.2.5

Eliminate the common factors.

Step:3.2.5.1

Cancel out the common $x$ terms.

Step:3.2.5.2

Simplify the expression to $1+\ln (x)+\frac{d}{dx}(-x)$.

Step:3.2.6

Multiply $\ln (x)$ by $1$ to get $1+\ln (x)+\frac{d}{dx}(-x)$.

Step:3.3

Find the derivative of $-x$.

Step:3.3.1

Since $-1$ is a constant, the derivative of $-x$ is $-1\cdot \frac{d}{dx}(x)$.

Step:3.3.2

Use the Power Rule with $n=1$.

Step:3.3.3

Multiply $-1$ by $1$ to get $1+\ln (x)-1$.

Step:3.4

Combine like terms.

Step:3.4.1

Subtract $1$ from $1$ to get $0+\ln (x)$.

Step:3.4.2

Combine $0$ and $\ln (x)$ to get $\ln (x)$.

Step:4

Write the derivative of $y$ as equal to the simplified right-hand side: $\frac{dy}{dx}=\ln (x)$.

Step:5

Substitute $\frac{dy}{dx}$ for $y$ in the final expression: $\frac{dy}{dx}=\ln (x)$.

Knowledge Notes:

1. Derivative: The derivative of a function measures how the function value changes as its input changes. The notation $\frac{d}{dx}$ is used to denote the derivative with respect to $x$.

2. Sum Rule: The derivative of a sum of functions is the sum of their derivatives.

3. Product Rule: For two functions $u(x)$ and $v(x)$, the derivative of their product $uv$ is given by $u^{\prime }v+u{v}^{\prime }$.

4. Power Rule: The derivative of $x^n$ with respect to $x$ is $n{x}^{n-1}$.

5. Natural Logarithm: The natural logarithm function, denoted as $\ln (x)$, is the inverse of the exponential function $e^x$. The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$.

6. Combining Like Terms: Simplifying expressions by adding or subtracting like terms, which are terms that have the same variables raised to the same power.

7. Simplification: The process of reducing an expression to its simplest form by performing arithmetic operations and combining like terms.