Textbook Question
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
x = e^y; (2, ln 2)
4. Applications of Derivatives
Implicit Differentiation
3:12 minutesProblem 3.8.27
Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
sin x+sin y=y
Verified step by step guidance1
Start by differentiating both sides of the equation with respect to x. The equation is sin(x) + sin(y) = y.
Differentiate sin(x) with respect to x, which gives cos(x).
Differentiate sin(y) with respect to x. Since y is a function of x, use the chain rule: the derivative is cos(y) * (dy/dx).
Differentiate y with respect to x, which is simply dy/dx.
Combine all the differentiated parts: cos(x) + cos(y) * (dy/dx) = dy/dx. Solve this equation for dy/dx to find the derivative.
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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. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule when necessary. This method is particularly useful for equations involving both x and y, allowing us to find the derivative of y with respect to x.
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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 defined as a function of u, which in turn is a function of x, 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 rule is essential in implicit differentiation, especially when differentiating terms involving y.
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental functions in calculus that relate angles to ratios of sides in right triangles. In the context of the given equation, sin x and sin y represent the sine of the angles x and y, respectively. Understanding the properties and derivatives of these functions is crucial for applying implicit differentiation, as their derivatives (e.g., the derivative of sin y is cos y dy/dx) will be used in the differentiation process.
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Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
y = xe^y
Textbook Question
Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.