Find dy/dx at (-1,8) y=3x^2-5x ; (-1,8)

Solution:

Step 1:

Take the derivative of both sides of the equation with respect to $x$: $\frac{d}{dx}(y)=\frac{d}{dx}(3x^2-5x)$.

Step 2:

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

Step 3:

Apply differentiation to the right-hand side of the equation.

Step 3.1:

Utilize the Sum Rule in differentiation: $\frac{d}{dx}(3x^2)+\frac{d}{dx}(-5x)$.

Step 3.2:

Find the derivative of $3x^2$ with respect to $x$.

Step 3.2.1:

As $3$ is a constant, it remains unchanged during differentiation: $3\frac{d}{dx}(x^2)$.

Step 3.2.2:

Apply the Power Rule, which states that the derivative of $x^n$ is $n{x}^{n-1}$, where $n=2$: $3(2x)$.

Step 3.2.3:

Calculate $2\times 3$: $6x$.

Step 3.3:

Find the derivative of $-5x$ with respect to $x$.

Step 3.3.1:

As $-5$ is a constant, it remains unchanged during differentiation: $-5\frac{d}{dx}(x)$.

Step 3.3.2:

Apply the Power Rule, where $n=1$: $-5\cdot 1$.

Step 3.3.3:

Calculate $-5\times 1$: $-5$.

Step 4:

Combine the results to form the derivative equation: $\frac{dy}{dx}=6x-5$.

Step 5:

Substitute $\frac{dy}{dx}$ for $y$ in the equation: $\frac{dy}{dx}=6x-5$.

Step 6:

Plug in the values of $x=-1$ and $y=8$ into the derivative equation: $\frac{dy}{dx}=6(-1)-5$.

Step 7:

Simplify the expression to find the derivative at the given point.

Step 7.1:

Perform the multiplication: $-6-5$.

Step 7.2:

Combine the terms to get the final answer: $\frac{dy}{dx}=-11$ at the point $(-1,8)$.

Knowledge Notes:

1. Derivative: The derivative of a function at a point is the rate at which the function's value changes at that point. It is a fundamental concept in calculus.

2. Sum Rule: This rule states that the derivative of a sum of functions is the sum of the derivatives of those functions.

3. Power Rule: A basic differentiation rule that says if $f(x)=x^n$, then $f^{\prime }(x)=n{x}^{n-1}$.

4. Differentiation of Constants: Constants differentiate to zero, and when they multiply a function, they remain as coefficients in the derivative.

5. Substitution: After finding the derivative of a function, you can evaluate it at a specific point by substituting the values of the variables into the derivative.

6. Simplification: After substitution, algebraic simplification may be necessary to arrive at the final value of the derivative at the given point.