Find dy/dx y=2xe^x-2e^x

Solution:

Step 1: Differentiate the equation with respect to $x$.

$\frac{d}{dx}(y) = \frac{d}{dx}(2xe^x - 2e^x)$

Step 2: The derivative of $y$ with respect to $x$ is denoted by $\frac{dy}{dx}$.

$\frac{dy}{dx}$

Step 3: Apply differentiation rules to the right-hand side of the equation.

Step 3.1: Utilize the Sum Rule to differentiate each term separately.

$\frac{d}{dx}(2xe^x) + \frac{d}{dx}(-2e^x)$

Step 3.2: Differentiate $2xe^x$.

Step 3.2.1: Treat the constant $2$ and apply the derivative to the remaining function.

$2\frac{d}{dx}(xe^x)$

Step 3.2.2: Apply the Product Rule to $xe^x$.

$2(x\frac{d}{dx}(e^x) + e^x\frac{d}{dx}(x))$

Step 3.2.3: Apply the Exponential Rule to $e^x$.

$2(xe^x + e^x\frac{d}{dx}(x))$

Step 3.2.4: Apply the Power Rule to $x$.

$2(xe^x + e^x \cdot 1)$

Step 3.2.5: Simplify the expression.

$2(xe^x + e^x)$

Step 3.3: Differentiate $-2e^x$.

Step 3.3.1: Treat the constant $-2$ and apply the derivative to $e^x$.

$-2\frac{d}{dx}(e^x)$

Step 3.3.2: Apply the Exponential Rule to $e^x$.

$-2e^x$

Step 3.4: Combine and simplify the terms.

Step 3.4.1: Distribute the constant $2$.

$2xe^x + 2e^x - 2e^x$

Step 3.4.2: Combine like terms.

$2xe^x$

Step 4: Write the final derivative expression.

$\frac{dy}{dx} = 2xe^x$

Knowledge Notes:

To solve this problem, we need to apply several rules of differentiation:

1. Sum Rule: The derivative of a sum of functions is the sum of the derivatives of each function. In mathematical terms, if $f(x) = g(x) + h(x)$, then $\frac{d}{dx}f(x) = \frac{d}{dx}g(x) + \frac{d}{dx}h(x)$.

2. Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Formally, if $f(x) = c \cdot g(x)$ where $c$ is a constant, then $\frac{d}{dx}f(x) = c \cdot \frac{d}{dx}g(x)$.

3. Product Rule: The derivative of the product of two functions is given by $f'(x)g(x) + f(x)g'(x)$. If $f(x) = g(x) \cdot h(x)$, then $\frac{d}{dx}f(x) = g'(x)h(x) + g(x)h'(x)$.

4. Exponential Rule: The derivative of an exponential function with base $e$ is the original function itself. If $f(x) = e^x$, then $\frac{d}{dx}f(x) = e^x$.

5. Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$. If $f(x) = x^n$, then $\frac{d}{dx}f(x) = nx^{n-1}$.

In this problem, we apply these rules to differentiate the function $y = 2xe^x - 2e^x$ with respect to $x$. The solution involves applying the Sum Rule, Constant Multiple Rule, Product Rule, Exponential Rule, and Power Rule in sequence to find the derivative $\frac{dy}{dx}$. The final result is $\frac{dy}{dx} = 2xe^x$, which is the rate of change of $y$ with respect to $x$.