Finding the Derivative of the Equation

Oct 13, 2025 by leolavanderia.com.br 141 views

A Point Moving Along a Graph: Calculating the Rate of Change

In calculus, we often encounter problems that involve rates of change. One common scenario is when a point moves along a graph, and we need to find the rate at which the coordinates of the point are changing. In this article, we will explore how to calculate the rate of change of the y-coordinate of a point moving along the graph of a given equation.

The Equation of the Graph

The equation of the graph is given by ${$ y = x + x^2 + 7 $}$. This is a quadratic equation, and its graph is a parabola that opens upwards. The point ${$ P(x, y) $}$ moves along this graph, and we are interested in finding the rate at which the y-coordinate is changing.

The Rate of Change of the x-Coordinate

We are given that the x-values are changing at a rate of 4 units per second. This means that the derivative of the x-coordinate with respect to time is ${$ \frac{dx}{dt} = 4 $}$.

The Chain Rule

To find the rate of change of the y-coordinate, we need to use the chain rule. The chain rule states that if we have a composite function of the form ${$ y = f(g(x)) $}$, then the derivative of y with respect to x is given by ${$ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) $}$.

In our case, we have ${$ y = x + x^2 + 7 $}$, and we want to find the rate of change of y with respect to time. We can rewrite the equation as ${$ y = f(x) = x + x^2 + 7 $}$, where ${$ f(x) $}$ is a composite function.

Finding the Derivative of the Equation

To find the derivative of the equation, we need to find the derivative of the outer function ${$ f(x) = x + x^2 + 7 $}$ with respect to x, and then multiply it by the derivative of the inner function ${$ g(x) = x $}$ with respect to x.

The derivative of the outer function is ${$ f'(x) = 1 + 2x $}$.

The derivative of the inner function is ${$ g'(x) = 1 $}$.

Applying the Chain Rule

Now we can apply the chain rule to find the rate of change of y with respect to time. We have ${$ \frac{dy}{dt} = f'(g(x)) \cdot g'(x) $}$.

Substituting the values, we get ${$ \frac{dy}{dt} = (1 + 2x) \cdot 1 $}$.

Finding the Rate of Change at a Given Point

We are interested in finding the rate of change of y at the point ${$ Q(1, 9) $}$. To do this, we need to substitute x = 1 into the equation ${$ \frac{dy}{dt} = (1 + 2x) \cdot 1 $}$.

Substituting x = 1, we get ${$ \frac{dy}{dt} = (1 + 2(1)) \cdot 1 = 3 $}$.

A point moves along the graph of the equation ${$ y = 2x^2 + 3x - 4 $}$. The x-values are changing at a rate of 2 units per second. Find the rate at which the y-values are changing at the point ${$ P(1, 2) $}$.
A point moves along the graph of the equation ${$ y = x^3 - 2x^2 + x + 1 $}$. The x-values are changing at a rate of 3 units per second. Find the rate at which the y-values are changing at the point ${$ Q(2, 5) $}$.

To solve the first example problem, we need to find the derivative of the equation ${$ y = 2x^2 + 3x - 4 $}$ with respect to x. The derivative is ${$ \frac{dy}{dx} = 4x + 3 $}$. We can then apply the chain rule to find the rate of change of y with respect to time. We have ${$ \frac{dy}{dt} = (4x + 3) \cdot 2 $}$. Substituting x = 1, we get ${$ \frac{dy}{dt} = (4(1) + 3) \cdot 2 = 14 $}$.
To solve the second example problem, we need to find the derivative of the equation ${$ y = x^3 - 2x^2 + x + 1 $}$ with respect to x. The derivative is ${$ \frac{dy}{dx} = 3x^2 - 4x + 1 $}$. We can then apply the chain rule to find the rate of change of y with respect to time. We have ${$ \frac{dy}{dt} = (3x^2 - 4x + 1) \cdot 3 $}$. Substituting x = 2, we get ${$ \frac{dy}{dt} = (3(2)^2 - 4(2) + 1) \cdot 3 = 27 $}$.

In this article, we have seen how to calculate the rate of change of the y-coordinate of a point moving along the graph of a given equation. We used the chain rule to find the derivative of the equation, and then applied it to find the rate of change of y with respect to time. We also found the rate of change of y at a given point, ${$ Q(1, 9) $}$. We also solved two example problems to demonstrate the application of the chain rule in finding the rate of change of y with respect to time.
A Point Moving Along a Graph: Calculating the Rate of Change - Q&A

In our previous article, we explored how to calculate the rate of change of the y-coordinate of a point moving along the graph of a given equation. We used the chain rule to find the derivative of the equation, and then applied it to find the rate of change of y with respect to time. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the chain rule, and how is it used to find the rate of change of y with respect to time?

A: The chain rule is a mathematical concept that allows us to find the derivative of a composite function. In the context of finding the rate of change of y with respect to time, the chain rule is used to find the derivative of the equation with respect to x, and then multiply it by the derivative of the inner function with respect to x.

Q: How do I find the derivative of the equation with respect to x?

A: To find the derivative of the equation with respect to x, you need to apply the power rule of differentiation. The power rule states that if y = x^n, then y' = nx^(n-1).

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a mathematical concept that allows us to find the derivative of a function of the form y = x^n. The power rule states that if y = x^n, then y' = nx^(n-1).

Q: How do I apply the chain rule to find the rate of change of y with respect to time?

A: To apply the chain rule, you need to find the derivative of the equation with respect to x, and then multiply it by the derivative of the inner function with respect to x. The formula for the chain rule is:

dy/dt = (dy/dx) * (dx/dt)

Q: What is the formula for the chain rule?

A: The formula for the chain rule is:

dy/dt = (dy/dx) * (dx/dt)

Q: How do I find the rate of change of y with respect to time at a given point?

A: To find the rate of change of y with respect to time at a given point, you need to substitute the x-value of the point into the equation for the derivative of the equation with respect to x, and then multiply it by the derivative of the inner function with respect to x.

Q: What is the significance of the chain rule in calculus?

A: The chain rule is a fundamental concept in calculus that allows us to find the derivative of composite functions. It is used extensively in physics, engineering, and economics to model real-world phenomena.

Q: Can the chain rule be used to find the rate of change of y with respect to time in other contexts?

A: Yes, the chain rule can be used to find the rate of change of y with respect to time in other contexts, such as in the study of population growth, chemical reactions, and electrical circuits.

In this article, we have answered some frequently asked questions related to the topic of calculating the rate of change of the y-coordinate of a point moving along the graph of a given equation. We have also provided a brief overview of the chain rule and its significance in calculus. We hope that this article has been helpful in clarifying any doubts you may have had about this topic.

A point moves along the graph of the equation ${$ y = 2x^2 + 3x - 4 $}$. The x-values are changing at a rate of 2 units per second. Find the rate at which the y-values are changing at the point ${$ P(1, 2) $}$.
A point moves along the graph of the equation ${$ y = x^3 - 2x^2 + x + 1 $}$. The x-values are changing at a rate of 3 units per second. Find the rate at which the y-values are changing at the point ${$ Q(2, 5) $}$.

To solve the first example problem, we need to find the derivative of the equation ${$ y = 2x^2 + 3x - 4 $}$ with respect to x. The derivative is ${$ \frac{dy}{dx} = 4x + 3 $}$. We can then apply the chain rule to find the rate of change of y with respect to time. We have ${$ \frac{dy}{dt} = (4x + 3) \cdot 2 $}$. Substituting x = 1, we get ${$ \frac{dy}{dt} = (4(1) + 3) \cdot 2 = 14 $}$.
To solve the second example problem, we need to find the derivative of the equation ${$ y = x^3 - 2x^2 + x + 1 $}$ with respect to x. The derivative is ${$ \frac{dy}{dx} = 3x^2 - 4x + 1 $}$. We can then apply the chain rule to find the rate of change of y with respect to time. We have ${$ \frac{dy}{dt} = (3x^2 - 4x + 1) \cdot 3 $}$. Substituting x = 2, we get ${$ \frac{dy}{dt} = (3(2)^2 - 4(2) + 1) \cdot 3 = 27 $}$.