Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.8.9

Chapter 3, Problem 3.8.9

If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.

Verified step by step guidance

First, recognize that L is a function of x and y, specifically L = √(x² + y²). This is the formula for the length of the hypotenuse in a right triangle with legs x and y.

To find dL/dt, use implicit differentiation with respect to time t. Start by differentiating both sides of the equation L = √(x² + y²) with respect to t.

Apply the chain rule to differentiate L = √(x² + y²). The derivative of L with respect to t is (1/2) * (1/√(x² + y²)) * (2x * dx/dt + 2y * dy/dt).

Substitute the given values into the differentiated equation: dx/dt = -1, dy/dt = 3, x = 5, and y = 12.

Simplify the expression to find dL/dt. Remember to substitute the values into the expression and simplify step by step, ensuring each part of the equation is correctly calculated.

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Key Concepts

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

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a variable z depends on y, which in turn depends on x, then the derivative of z with respect to x can be found by multiplying the derivative of z with respect to y by the derivative of y with respect to x. In this problem, it helps in finding dL/dt by relating the derivatives of L with respect to x and y.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. It is particularly useful when dealing with equations involving multiple variables. In this scenario, implicit differentiation allows us to differentiate L = √(x² + y²) with respect to time, t, by considering the derivatives dx/dt and dy/dt.

Pythagorean Theorem

The Pythagorean theorem is a geometric principle that relates the lengths of the sides of a right triangle. It states that the square of the hypotenuse (L) is equal to the sum of the squares of the other two sides (x and y). In this problem, L = √(x² + y²) represents the hypotenuse, and understanding this relationship is crucial for setting up the equation to find dL/dt.

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