Find dy/dx x^3-y^3=1/2
The problem provided is asking for the derivative of y with respect to x, denoted as dy/dx, from the given equation x^3 - y^3 = 1/2. Essentially, you are tasked with finding the rate at which y changes as x changes, given the relationship between x and y established by the equation. This problem involves using implicit differentiation because y is defined implicitly in terms of x by the equation, rather than being given as an explicit function y=f(x).
$x^{3} - y^{3} = \frac{1}{2}$
Answer
Solution:
Step:1
Apply differentiation to both sides of the given equation with respect to $x$:
$$\frac{d}{dx}(x^3 - y^3) = \frac{d}{dx}\left(\frac{1}{2}\right)$$
Step:2
Differentiate the left-hand side term by term.
Step:2.1
Perform differentiation.
Step:2.1.1
Utilize the Sum Rule in differentiation, which allows us to differentiate terms separately:
$$\frac{d}{dx}(x^3) - \frac{d}{dx}(y^3)$$
Step:2.1.2
Apply the Power Rule for differentiation, $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n=3$:
$$3x^2 - \frac{d}{dx}(y^3)$$
Step:2.2
Find the derivative of $-y^3$ with respect to $x$.
Step:2.2.1
Extract the constant factor $-1$ and differentiate $y^3$:
$$3x^2 - \frac{d}{dx}(y^3)$$
Step:2.2.2
Apply the Chain Rule for differentiation, $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$, where $f(u) = u^3$ and $g(x) = y$.
Step:2.2.2.1
Let $u = y$ to prepare for the Chain Rule:
$$3x^2 - \left(\frac{d}{du}(u^3) \frac{dy}{dx}\right)$$
Step:2.2.2.2
Differentiate $u^3$ using the Power Rule:
$$3x^2 - (3u^2 \frac{dy}{dx})$$
Step:2.2.2.3
Substitute $u$ back with $y$:
$$3x^2 - (3y^2 \frac{dy}{dx})$$
Step:2.2.3
Express $\frac{dy}{dx}$ as $y'$:
$$3x^2 - (3y^2 y')$$
Step:2.2.4
Combine the terms and simplify:
$$3x^2 - 3y^2 y'$$
Step:3
The derivative of a constant, $\frac{1}{2}$, with respect to $x$ is zero:
$$0$$
Step:4
Combine the differentiated left and right sides to form the equation:
$$3x^2 - 3y^2 y' = 0$$
Step:5
Isolate $y'$ (the derivative of $y$ with respect to $x$).
Step:5.1
Move $3x^2$ to the right side of the equation:
$$-3y^2 y' = -3x^2$$
Step:5.2
Divide both sides of the equation by $-3y^2$ to solve for $y'$.
Step:5.2.1
Perform the division:
$$\frac{-3y^2 y'}{-3y^2} = \frac{-3x^2}{-3y^2}$$
Step:5.2.2
Simplify the left side:
Step:5.2.2.1
Eliminate the common factor of $-3$:
$$\frac{y^2 y'}{y^2} = \frac{-3x^2}{-3y^2}$$
Step:5.2.2.2
Reduce the $y^2$ terms:
$$y' = \frac{-3x^2}{-3y^2}$$
Step:5.2.3
Simplify the right side:
Step:5.2.3.1
Eliminate the common factor of $-3$:
$$y' = \frac{x^2}{y^2}$$
Step:6
Replace $y'$ with $\frac{dy}{dx}$ to get the final derivative:
$$\frac{dy}{dx} = \frac{x^2}{y^2}$$
Knowledge Notes:
Sum Rule: This rule states that the derivative of a sum of functions is the sum of their derivatives. For functions $f(x)$ and $g(x)$, the rule is expressed as $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$.
Power Rule: A fundamental rule in differentiation, which states that for any real number $n$, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$. Formally, $\frac{d}{dx}(x^n) = nx^{n-1}$.
Chain Rule: This rule is used for differentiating compositions of functions. If $y = f(u)$ and $u = g(x)$, then the derivative of $y$ with respect to $x$ is given by $\frac{dy}{dx} = \frac{df}{du} \cdot \frac{du}{dx}$.
Differentiating Constants: The derivative of a constant is always zero because constants do not change as the variable changes.
Implicit Differentiation: This technique is used when it is difficult or impossible to solve for $y$ explicitly in terms of $x$. Instead, we differentiate both sides of the equation with respect to $x$, treating $y$ as an implicit function of $x$.
