Derivatives in Differential Form

In Exercises 17–28, find dy.

xy² − 4x³/² − y = 0

Verified step by step guidance

First, identify the given equation: \( xy^2 - 4x^{3/2} - y = 0 \). This is an implicit function involving both x and y.

To find \( \frac{dy}{dx} \), we need to differentiate both sides of the equation with respect to x. Remember that y is a function of x, so use implicit differentiation.

Differentiate each term: For \( xy^2 \), use the product rule: \( \frac{d}{dx}(xy^2) = x \frac{d}{dx}(y^2) + y^2 \frac{d}{dx}(x) \). For \( y^2 \), use the chain rule: \( \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \).

Differentiate \( -4x^{3/2} \) using the power rule: \( \frac{d}{dx}(-4x^{3/2}) = -4 \cdot \frac{3}{2}x^{1/2} \).

Differentiate \( -y \) with respect to x: \( \frac{d}{dx}(-y) = -\frac{dy}{dx} \). Combine all differentiated terms and solve for \( \frac{dy}{dx} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this method, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule when necessary. This allows us to find the derivative of one variable in terms of the other, even when they are intertwined.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is crucial when dealing with implicit functions where one variable depends on another.

Differential Notation

Differential notation involves expressing derivatives in terms of differentials, such as dy and dx. In the context of implicit differentiation, dy represents the change in the dependent variable y, while dx represents the change in the independent variable x. Understanding this notation is essential for solving equations involving derivatives, as it helps clarify the relationship between the variables and their rates of change.

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Textbook Question

Parallel tangent lines Find the two points where the curve x² + xy + y² = 7 crosses the x-axis, and show that the tangent lines to the curve at these points are parallel. What is the common slope of these tangent lines?

Textbook Question

Finding Derivative Functions and Values

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

f(x) = 4 – x²; f′(−3), f′(0), f′(1)

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Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = (2√x)/(3(1 + √x))

Textbook Question

In Exercises 19–22, find the slope of the curve at the point indicated.

y = x³ − 2x + 7, x = −2