Find dy/dx y^2=2x
The problem provided is asking for the derivative of the function y with respect to x, denoted as dy/dx. It involves a given equation y^2 = 2x, which represents a relationship between the variables y and x. The task is to find the rate at which y changes with respect to x by differentiating this implicit equation. This involves using implicit differentiation, a technique used when a function is not given in the standard form y = f(x), but rather in a form where y and x are intermingled.
$y^{2} = 2 x$
Answer
Solution:
Step:1 Take the derivative of both sides with respect to $x$. $\frac{d}{dx}(y^2) = \frac{d}{dx}(2x)$
Step:2 Apply the derivative to the left-hand side.
Step:2.1 Utilize the chain rule for differentiation, which is $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, where $f(x) = x^2$ and $g(x) = y$.
Step:2.1.1 Introduce $u = y$ to apply the chain rule. $\frac{d}{du}(u^2) \frac{dy}{dx}$
Step:2.1.2 Apply the power rule, which states $\frac{d}{du}(u^n) = nu^{n-1}$, where $n = 2$. $2u \frac{dy}{dx}$
Step:2.1.3 Substitute $y$ back in for $u$. $2y \frac{dy}{dx}$
Step:2.2 Express $\frac{dy}{dx}$ as $dy/dx$. $2y \cdot dy/dx$
Step:3 Differentiate the right-hand side.
Step:3.1 Recognize that $2$ is a constant and differentiate $2x$ with respect to $x$. $2 \frac{d}{dx}(x)$
Step:3.2 Apply the power rule to $x$, where $n = 1$. $2 \cdot 1$
Step:3.3 Simplify the multiplication of $2$ by $1$. $2$
Step:4 Combine the differentiated left and right sides into an equation. $2y \cdot dy/dx = 2$
Step:5 Isolate $dy/dx$ by dividing both sides by $2y$.
Step:5.1 Divide both sides by $2y$. $\frac{2y \cdot dy/dx}{2y} = \frac{2}{2y}$
Step:5.2 Simplify the left-hand side.
Step:5.2.1 Reduce the common factor of $2$. $\frac{\cancel{2}y \cdot dy/dx}{\cancel{2}y} = \frac{2}{2y}$
Step:5.2.1.1 Simplify the expression. $\frac{dy/dx}{1} = \frac{2}{2y}$
Step:5.2.2 Reduce the common factor of $y$. $\frac{dy/dx}{1} = \frac{2}{2y}$
Step:5.2.2.1 Simplify the expression. $dy/dx = \frac{2}{2y}$
Step:5.3 Simplify the right-hand side.
Step:5.3.1 Reduce the common factor of $2$. $dy/dx = \frac{\cancel{2}}{2 \cancel{y}}$
Step:5.3.1.1 Simplify the expression. $dy/dx = \frac{1}{y}$
Step:6 Finalize the expression for $\frac{dy}{dx}$. $\frac{dy}{dx} = \frac{1}{y}$
Knowledge Notes:
The problem-solving process involves finding the derivative of a function with respect to $x$, denoted as $\frac{dy}{dx}$, where $y^2 = 2x$. The steps include:
Differentiation: Applying the derivative operation to both sides of the equation with respect to $x$.
Chain Rule: A rule in calculus for differentiating the composition of two or more functions. It states that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$.
Power Rule: A basic rule of differentiation that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Simplification: The process of reducing an expression to its simplest form by canceling out common factors and combining like terms.
Isolation of Variable: The process of rearranging an equation to solve for a particular variable, in this case, isolating $\frac{dy}{dx}$.
In this problem, the chain rule is used because $y$ is a function of $x$, even though it is not explicitly given. The power rule is applied to both $y^2$ and $2x$ because they are both power functions. The simplification steps involve algebraic manipulation to isolate $\frac{dy}{dx}$ on one side of the equation. The final result gives the derivative of $y$ with respect to $x$.
