Find dy/dx at (-1,8) y=3x^2-5x ; (-1,8)
The problem is asking to perform differentiation with respect to x, on the given function y = 3x^2 - 5x, and then evaluate the derivative at the specific point (-1, 8). To do this, one must apply the rules of differentiation to find the general expression for the derivative dy/dx, which represents the slope of the tangent line to the curve at any point (x, y). After finding this general expression, the x-coordinate of the given point (-1, 8) will be substituted into it to find the slope of the tangent at that particular point.
$y = 3 x^{2} - 5 x$;$\left(\right. - 1 , 8 \left.\right)$
Answer
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 $nx^{n-1}$, 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:
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.
Sum Rule: This rule states that the derivative of a sum of functions is the sum of the derivatives of those functions.
Power Rule: A basic differentiation rule that says if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Differentiation of Constants: Constants differentiate to zero, and when they multiply a function, they remain as coefficients in the derivative.
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.
Simplification: After substitution, algebraic simplification may be necessary to arrive at the final value of the derivative at the given point.
