Question

Carry out the following steps. a. Use implicit differentiation to find \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. $$x y=7 ;(1,7)$$

Short answer

Answer: The slope of the curve at the point \((1, 7)\) is \(-7\).

Step-by-step solution

Rewrite the equation with proper notation

Write down the given equation: \(xy = 7\). Notice that the equation is already expressed implicitly, so we can proceed to the next step.

Apply implicit differentiation

To find the derivative of y with respect to x, \(\frac{dy}{dx}\), apply implicit differentiation to both sides of the equation. Differentiating both sides with respect to x, we get: \(\frac{d}{dx}(x\cdot y) = \frac{d}{dx}(7)\)

Use product rule

We need to differentiate \(xy\) using the product rule, which states: \(\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}\). In our case, \(u = x\) and \(v = y\). Applying the product rule, we get: \(x \frac{dy}{dx} + y\frac{dx}{dx} = 0\)

Simplify and solve for \(\frac{dy}{dx}\)

Simplify the expression: \(x \frac{dy}{dx} + y(1) = 0\) Now, solve for \(\frac{dy}{dx}\): \(\frac{dy}{dx} = -\frac{y}{x}\)

Plug in the coordinates of the given point

Substitute the given coordinates \((1, 7)\) into the derived expression for \(\frac{dy}{dx}\) to find the slope of the curve at that point. \(\frac{dy}{dx}\Big|_{(1,7)} = -\frac{7}{1} = -7\) The slope of the curve at the given point \((1, 7)\) is \(-7\).

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. In the context of our problem, it allows us to find the rate at which one variable changes relative to another, which is known as the derivative. Implicit differentiation, a technique used in calculus, is particularly useful when dealing with equations where the variables cannot be easily separated.
In the given exercise, calculus is applied through implicit differentiation to find the derivative \( \frac{dy}{dx} \) in an equation where y is not isolated. This derivative signifies the rate of change of y with respect to x. Calculus not only helps to determine the instantaneous rate of change but also enables us to find the slope of a curve at a specific point, which is an invaluable tool for understanding the behavior of functions in various fields such as physics, engineering, and economics.

Product Rule

The product rule is a fundamental law in calculus used to find the derivative of a product of two functions. It states that if you have a product of two functions \(u(x)\) and \(v(x)\), their derivative \( \frac{d}{dx}(uv) \) can be found using the formula: \[\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}\].
In our exercise, we applied the product rule to differentiate \(xy\). The variable \(x\) was treated as \(u\) while \(y\) was treated as \(v\), resulting in the derivative \(x \frac{dy}{dx} + y \frac{dx}{dx}\). Understanding and applying the product rule is crucial because it allows us to differentiate more complex expressions accurately that are not merely sums or differences of functions but their products.

Slope of a Curve

The slope of a curve at a particular point is a measure of how steep the curve is at that point. It is the 'rise over run' for an infinitesimally small interval on the curve. In a graphical sense, the slope is the angle of the tangent line to the curve at that point. When you find the derivative of a curve at a specific point, you are essentially computing this slope.
For the exercise, after using implicit differentiation and the product rule, we found that the slope of the curve defined by the equation \(xy = 7\) at the point \( (1, 7) \) is \( -7 \). This tells us that at this point, if we move one unit to the right along the x-axis, the curve will drop by seven units along the y-axis. The negative sign indicates that the curve is decreasing at that point. The concept of slope is a key element in many applications of calculus, for example, in optimizing functions and solving physics problems related to motion.