Find dy/dx y=x^2-2x-3

Solution:

Step:1 Apply differentiation to each term in the equation $y=x^2-2x-3$ to find $\frac{dy}{dx}$.

Step:2 Recognize that the derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.

Step:3 Proceed to differentiate the expression on the right side term by term.

Step:3.1 Begin differentiation.

Step:3.1.1 Utilize the Sum Rule of differentiation, which allows us to differentiate each term separately: $\frac{d}{dx}(x^2)+\frac{d}{dx}(-2x)+\frac{d}{dx}(-3)$.

Step:3.1.2 Apply the Power Rule for differentiation, which states that the derivative of $x^n$ is $n{x}^{n-1}$, where $n$ is a constant, to the term $x^2$: $2x^{2-1}+\frac{d}{dx}(-2x)+\frac{d}{dx}(-3)$.

Step:3.2 Differentiate the term $-2x$.

Step:3.2.1 Considering $-2$ as a constant, we differentiate $-2x$ with respect to $x$: $2x-2\frac{d}{dx}(x)+\frac{d}{dx}(-3)$.

Step:3.2.2 Again, apply the Power Rule for the term $x$, where $n=1$: $2x-2(1)+\frac{d}{dx}(-3)$.

Step:3.2.3 Simplify the expression by multiplying $-2$ by $1$: $2x-2+\frac{d}{dx}(-3)$.

Step:3.3 Employ the Constant Rule of differentiation.

Step:3.3.1 Since $-3$ is a constant, its derivative with respect to $x$ is $0$: $2x-2+0$.

Step:3.3.2 Combine the terms $2x-2$ and $0$ to simplify: $2x-2$.

Step:4 Express the derivative of $y$ with respect to $x$ by equating the left side to the differentiated right side: $\frac{dy}{dx}=2x-2$.

Step:5 Substitute $\frac{dy}{dx}$ for $y$ in the final expression: $\frac{dy}{dx}=2x-2$.

Knowledge Notes:

1. Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. It's represented by the derivative of the function.

2. The Sum Rule of differentiation states that the derivative of a sum of functions is the sum of the derivatives of those functions.

3. The Power Rule is a basic differentiation rule that says if $f(x)=x^n$ for a real number $n$, then $f^{\prime }(x)=n{x}^{n-1}$.

4. The Constant Rule of differentiation states that the derivative of a constant is zero.

5. In the context of differentiation, a constant is a value that does not depend on the variable and therefore has a derivative of zero.

6. The notation $\frac{dy}{dx}$ represents the derivative of $y$ with respect to $x$, indicating how $y$ changes in response to changes in $x$.