Find dy/dx y^2=3x
The problem presented here involves calculus, specifically differentiation. The question is asking you to find the derivative of y with respect to x, denoted as dy/dx, for the given equation y^2 = 3x. This entails applying the rules of differentiation to both sides of the equation with respect to x, to solve for dy/dx. This would typically involve using implicit differentiation, since y is raised to a power and is not isolated. Implicit differentiation is used when dealing with equations where y and x are not easily separable.
$y^{2} = 3 x$
Answer
Solution:
Step 1:
Apply differentiation to both sides of the equation with respect to $x$: $\frac{d}{dx}(y^2) = \frac{d}{dx}(3x)$.
Step 2:
Differentiate the left-hand side using the chain rule.
Step 2.1:
Invoke the chain rule for differentiation: $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$, where $f(x) = x^2$ and $g(x) = y$.
Step 2.1.1:
Set $u = y$ to facilitate the application of the chain rule: $\frac{d}{du}(u^2) \cdot \frac{dy}{dx}$.
Step 2.1.2:
Apply the power rule for differentiation, which states $\frac{d}{du}(u^n) = n \cdot u^{n-1}$ for $n = 2$: $2u \cdot \frac{dy}{dx}$.
Step 2.1.3:
Substitute $y$ back in for $u$: $2y \cdot \frac{dy}{dx}$.
Step 2.2:
Express $\frac{dy}{dx}$ as $y'$: $2y \cdot y'$.
Step 3:
Differentiate the right-hand side of the equation.
Step 3.1:
Recognize that $3$ is a constant and differentiate $3x$ with respect to $x$: $3 \cdot \frac{d}{dx}(x)$.
Step 3.2:
Apply the power rule to $x$, where $n = 1$: $3 \cdot 1$.
Step 3.3:
Multiply $3$ by $1$ to get $3$.
Step 4:
Combine the differentiated left and right sides: $2y \cdot y' = 3$.
Step 5:
Isolate $y'$ by dividing both sides of the equation by $2y$.
Step 5.1:
Divide both sides by $2y$: $\frac{2y \cdot y'}{2y} = \frac{3}{2y}$.
Step 5.2:
Simplify the left-hand side.
Step 5.2.1:
Eliminate the common factor of $2$.
Step 5.2.1.1:
Cancel out the $2$s: $\frac{\cancel{2}y \cdot y'}{\cancel{2}y} = \frac{3}{2y}$.
Step 5.2.1.2:
Rewrite the simplified expression: $\frac{y \cdot y'}{y} = \frac{3}{2y}$.
Step 5.2.2:
Eliminate the common factor of $y$.
Step 5.2.2.1:
Cancel out the $y$s: $\frac{\cancel{y} \cdot y'}{\cancel{y}} = \frac{3}{2y}$.
Step 5.2.2.2:
Simplify to find $y'$: $y' = \frac{3}{2y}$.
Step 6:
Substitute $y'$ with $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{3}{2y}$.
Knowledge Notes:
The problem involves finding the derivative of $y$ with respect to $x$ given the equation $y^2 = 3x$. The solution employs several fundamental concepts of calculus:
Differentiation: The process of finding the derivative, which represents the rate at which a function is changing at any given point.
Chain Rule: A rule for differentiating compositions of functions. It states that the derivative of $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.
Power Rule: A basic rule of differentiation that states if $f(x) = x^n$, then $f'(x) = n \cdot x^{n-1}$.
Constants in Differentiation: Constants are differentiated to zero. When differentiating a constant multiplied by a function, the constant remains and the function is differentiated.
Simplifying Expressions: After differentiating, it is often necessary to simplify the expression to isolate the derivative ($\frac{dy}{dx}$ or $y'$).
In this problem, the chain rule is used to differentiate $y^2$ because $y$ is a function of $x$. The power rule is applied to both $y^2$ and $3x$. After differentiation, algebraic manipulation is used to solve for $\frac{dy}{dx}$.
