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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.8.31
Chapter 3, Problem 3.8.31

Use implicit differentiation to find dy/dx.
sin xy = x+y

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1
Start by differentiating both sides of the equation with respect to x. The equation is sin(xy) = x + y.
Apply the chain rule to differentiate sin(xy). The derivative of sin(u) with respect to u is cos(u), so differentiate sin(xy) to get cos(xy) * (d/dx)(xy).
Use the product rule to differentiate xy with respect to x. The product rule states that d(uv)/dx = u(dv/dx) + v(du/dx). Here, u = x and v = y, so the derivative is y + x(dy/dx).
Substitute the result from the product rule into the chain rule expression: cos(xy) * (y + x(dy/dx)).
Differentiate the right side of the equation, x + y, with respect to x. The derivative of x is 1, and the derivative of y with respect to x is dy/dx. Set the derivatives equal: cos(xy) * (y + x(dy/dx)) = 1 + dy/dx. Solve for 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. 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 as necessary. This method is particularly useful for equations involving products or compositions of functions.
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Finding The Implicit Derivative

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 is essential in implicit differentiation when dealing with products of variables.
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Intro to the Chain Rule

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 implicit differentiation, understanding how to differentiate these functions is crucial, as they often appear in equations involving angles and products of variables. For example, the derivative of sin(u) is cos(u) multiplied by the derivative of u, which is applied when differentiating equations like sin(xy).
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Introduction to Trigonometric Functions
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Find (f^−1)′(3), where f(x)=x³+x+1.

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Suppose f is a one-to-one function with f(2)=8 and f′(2)=4. What is the value of (f^−1)′(8)?

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Calculate the derivative of the following functions (i) using the fact that bx = exIn b and (ii) using logarithmic differentiation. Verify that both answers are the same.

y = (4x+1)In x

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Textbook Question

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.

x(1−y²)+y³=0

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Textbook Question

Derivatives Find and simplify the derivative of the following functions.

g(x) = e^x / x²-1

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