Find dy/dx y=x^2-2x-3
This question is asking for the derivative of the function y with respect to x, where y is defined as y = x^2 - 2x - 3. The derivative, denoted as dy/dx, represents the rate at which y changes with respect to x. Finding dy/dx involves applying differentiation rules to the polynomial function to calculate the slope of the tangent line to the curve at any given point on the graph of the function.
$y = x^{2} - 2 x - 3$
Answer
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 $nx^{n-1}$, 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:
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.
The Sum Rule of differentiation states that the derivative of a sum of functions is the sum of the derivatives of those functions.
The Power Rule is a basic differentiation rule that says if $f(x) = x^n$ for a real number $n$, then $f'(x) = nx^{n-1}$.
The Constant Rule of differentiation states that the derivative of a constant is zero.
In the context of differentiation, a constant is a value that does not depend on the variable and therefore has a derivative of zero.
The notation $\frac{dy}{dx}$ represents the derivative of $y$ with respect to $x$, indicating how $y$ changes in response to changes in $x$.
