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Chapter 3: Problem 2

Find \(d y / d x .\) Assume \(a, b, c\) are constants. $$x^{2}+y^{3}=8$$

Short Answer

Expert verified
\( \frac{dy}{dx} = \frac{-2x}{3y^2} \)

Step by step solution

01

Find the Derivative of Each Side

Start by differentiating each side with respect to x. On the left side, differentiate each term separately: \( \frac{d}{dx}(x^2 + y^3) = \frac{d}{dx}(x^2) + \frac{d}{dx}(y^3)\). On the right side, differentiate the constant 8 to get 0: \( \frac{d}{dx}(8) = 0 \).
02

Differentiate the First Term

Differentiate \(x^2\) with respect to \(x\). Since \( \frac{d}{dx}(x^2) = 2x \), the derivative of the first term is \(2x\).
03

Apply the Chain Rule to the Second Term

To differentiate \(y^3\) with respect to \(x\), use the chain rule: \( \frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx} \). This accounts for the fact that \(y\) is a function of \(x\).
04

Combine the Derivatives

Now, combine the derivatives from each term: \(2x + 3y^2 \frac{dy}{dx} = 0 \). This gives us the equation for the derivative of the left side.
05

Solve for \( \frac{dy}{dx} \)

Isolate \( \frac{dy}{dx} \) by subtracting \(2x\) from both sides, giving \( 3y^2 \frac{dy}{dx} = -2x \). Then, divide both sides by \(3y^2\) to solve for \( \frac{dy}{dx} \): \( \frac{dy}{dx} = \frac{-2x}{3y^2} \).

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Key Concepts

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

Chain Rule
Implicit differentiation often involves using the chain rule, a fundamental principle in calculus. The chain rule helps us find the derivative of a composite function. Imagine you have a function within a function. When y is a function of x, such as in the expression y3, the chain rule becomes vital.
Here's how it works:
  • Differentiate the outer function while keeping the inner function unchanged. For y3, this would be 3y2.
  • Multiply by the derivative of the inner function. Since y is a function of x, we multiply by dy/dx.
So, the derivative with respect to x becomes 3y2 dy/dx. Using the chain rule helps account for how changing x affects y.
This provides a complete view of the derivative concerning the related variable.
Derivative
The derivative is a key concept in calculus, representing the rate of change or slope of a function. When given an equation like \(x^2 + y^3 = 8\), finding the derivative dy/dx is like discovering how fast y changes as x changes.
  • For the term x2, the derivative with respect to x is straightforward, yielding 2x, showing a direct change.
  • The term y3, on the other hand, requires us to consider y as a function of x, introducing the concept of implicit differentiation.
In implicit differentiation, not all functions are solved for y. Instead, we differentiate each part in terms of x and solve for dy/dx. Differentiating implicitly is like peeling back layers to find the components affecting the rate of change.
Function of x
In calculus, understanding a function of x means recognizing the relationship between two variables, where one variable (usually y) depends on another (x). In the context of the equation \(x^2 + y^3 = 8\), thinking of y as a function of x means that y changes as x changes.
This concept is essential because it lays the groundwork for implicit differentiation.
  • When given a function or equation, identify which variables are dependent and independent.
  • Recognize when a variable might not be explicitly solved for, requiring implicit differentiation.
This approach helps manage complex relationships in equations and is crucial in applied mathematics where functions do not always appear in simple forms. When dealing with a function of x, it is how you explore interaction and dependency between variables, preparing you to find hidden or indirect rates of change.

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Most popular questions from this chapter

Explain what is wrong with the statement. The linear approximation for \(F(x)=x^{3}+1\) near \(x=0\) is an underestimate for the function \(F\) for all \(x, x \neq 0\)

Explain what is wrong with the statement.The derivative of \(f(x)=\sin (\sin x)\) is \(f^{\prime}(x)=\) \((\cos x)(\sin x)+(\sin x)(\cos x)\)

Let \(f\) be a differentiable function and let \(L\), be the linear function \(L(x)=f(a)+k(x-a)\) for some constant \(a\) Decide whether the following statements are true or false for all constants \(k .\) Explain your answer. (a) \(L\) is the local linearization for \(f\) near \(x=a\) (b) If \(\lim _{x \rightarrow a}(f(x)-L(x))=0,\) then \(L\) is the local linearization for \(f\) near \(x=a\).

Show that the power rule for derivatives applies to rational powers of the form \(y=x^{m / n}\) by raising both sides to the \(n^{\text {th }}\) power and using implicit differentiation.

Let \(f(x)=\ln (3 x)\) (a) Find \(f^{\prime}(x)\) and simplify your answer. (b) Use properties of logs to rewrite \(f(x)\) as a sum of logs. (c) Differentiate the result of part (b). Compare with the result in part (a).
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