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Chapter 3: Problem 14

Differentiate implicily to find \(d y / d x\). $$ x^{2}+y^{2}=25 $$

Short Answer

Expert verified
The derivative \( \frac{dy}{dx} \) is \( \frac{-x}{y} \).

Step by step solution

01

Differentiate both sides with respect to x

Differentiate each term on both sides of the equation with respect to x. Apply the chain rule where necessary.For the left-hand side:differentiate the first term: \[ \frac{d}{dx}(x^2) = 2x \]Now, differentiate the second term:\[ \frac{d}{dx}(y^2) = \frac{d}{dy}(y^2) \frac{dy}{dx} = 2y \frac{dy}{dx} \]Combine these results, and differentiate the right-hand side:\[ \frac{d}{dx}(25) = 0 \]
02

Combine the differentiated terms

Add the differentiated terms from both sides of the original equation:\[ 2x + 2y \frac{dy}{dx} = 0 \]
03

Solve for dy/dx

Isolate \( \frac{dy}{dx} \) on one side of the equation:Subtract \( 2x \) from both sides:\[ 2y \frac{dy}{dx} = -2x \]Divide both sides by \( 2y \):\[ \frac{dy}{dx} = \frac{-2x}{2y} = \frac{-x}{y} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is a branch of mathematics that studies how things change. It includes two main concepts: differentiation and integration. Differentiation helps us understand rates of change, while integration helps us find areas under curves and accumulations of quantities. In the problem given, we are using differentiation to find how changes in x affect changes in y. This is crucial for functions where both variables are interdependent, as is the case with implicit differentiation.
Chain Rule
Understanding the chain rule is key to solving implicit differentiation problems. The chain rule is a formula used to compute the derivative of a composite function. When differentiating a function like y with respect to x, we first differentiate y with respect to itself and then multiply by dy/dx. In our problem, we see the chain rule in action when we differentiate y虏:
  • First, we differentiate y虏 with respect to y, which gives us 2y.
  • Then, we multiply this result by dy/dx, reflecting how y changes with respect to x.
Combining these steps gives us:
  • dy/dx = (d/dy(y虏))(dy/dx) = 2y dy/dx.
Differentiation
Differentiation is the process of finding a derivative, which represents the rate at which a function is changing at any given point. In implicit differentiation, we find the derivative of both sides of an equation that defines y implicitly as a function of x. From our exercise, we differentiated each term with respect to x:
  • x虏 becomes 2x.
  • y虏 becomes 2y dy/dx using the chain rule.
We then combined these differentiated terms and solved for dy/dx, resulting in:
  • dy/dx = -x/y.
This tells us how y changes with respect to x under the given conditions.

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