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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 3.8.7
Chapter 3, Problem 3.8.7

5–8. Calculate dy/dx using implicit differentiation.
sin y+2 = x

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Start by understanding the equation: \( \sin y + 2 = x \). We need to find \( \frac{dy}{dx} \) using implicit differentiation.
Differentiate both sides of the equation with respect to \( x \). Remember that \( y \) is a function of \( x \), so when differentiating \( \sin y \), use the chain rule.
The derivative of \( \sin y \) with respect to \( x \) is \( \cos y \cdot \frac{dy}{dx} \). The derivative of the constant 2 is 0, and the derivative of \( x \) is 1.
Set up the equation from the differentiation: \( \cos y \cdot \frac{dy}{dx} = 1 \).
Solve for \( \frac{dy}{dx} \) by isolating it: \( \frac{dy}{dx} = \frac{1}{\cos y} \). This is the expression for the derivative using implicit differentiation.

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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 find the derivative of a function that is not explicitly solved for one variable in terms of another. Instead of isolating y, we differentiate both sides of the equation with respect to x, applying the chain rule when differentiating terms involving y. This method is particularly useful when dealing with equations where y cannot be easily expressed as a function of x.
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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 differentiating terms involving y.
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Intro to the Chain Rule

Trigonometric Derivatives

Trigonometric derivatives refer to the derivatives of trigonometric functions, which are essential in calculus. For example, the derivative of sin(y) with respect to x is cos(y) * (dy/dx) due to the chain rule. Understanding these derivatives is crucial when differentiating equations that involve trigonometric functions, as they help in simplifying the expressions and finding the required derivatives.
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Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question

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Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/4 cot x−1 / x−π/4

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Derivatives Find and simplify the derivative of the following functions.

g(t) = t³+3t²+t / t³

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Textbook Question

67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.


f(x) = x^2/3, for x>0

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State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?

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Textbook Question

Complete the following statement. If dy/dx is small, then small changes in x will result in relatively ______ changes in the value of y.