Find dy/dx y^2+6y+9x=0
The given question is asking for the derivative of the function y with respect to x, which is denoted as dy/dx. To be more specific, you are asked to find the first derivative of y in relation to x from the implicitly defined function y^2 + 6y + 9x = 0. This function is described implicitly because y is not isolated on one side of the equation; rather, y and x terms are mingled. The solution will require the application of implicit differentiation, a technique used when a function is not given in the form y = f(x), but can still be differentiated with respect to x.
$y^{2} + 6 y + 9 x = 0$
Answer
Solution:
Step 1:
Take the derivative of each term in the equation $y^2 + 6y + 9x = 0$ with respect to $x$.
$$\frac{d}{dx}(y^2 + 6y + 9x) = \frac{d}{dx}(0)$$
Step 2:
Apply the derivative to each term separately.
Step 2.1:
Utilize the Sum Rule to differentiate the sum term by term.
$$\frac{d}{dx}(y^2) + \frac{d}{dx}(6y) + \frac{d}{dx}(9x)$$
Step 2.2:
Find the derivative of $y^2$ with respect to $x$.
Step 2.2.1:
Employ the Chain Rule for differentiation, where $f(x) = x^2$ and $g(x) = y$.
$$\frac{d}{du}(u^2)\frac{dy}{dx} + \frac{d}{dx}(6y) + \frac{d}{dx}(9x)$$
Step 2.2.2:
Apply the Power Rule, where the derivative of $u^n$ is $nu^{n-1}$, with $n = 2$.
$$2u\frac{dy}{dx} + \frac{d}{dx}(6y) + \frac{d}{dx}(9x)$$
Step 2.2.3:
Substitute $u$ back with $y$.
$$2y\frac{dy}{dx} + \frac{d}{dx}(6y) + \frac{d}{dx}(9x)$$
Step 2.3:
Calculate the derivative of $6y$ with respect to $x$.
Step 2.3.1:
Since 6 is a constant, the derivative is $6\frac{dy}{dx}$.
$$2yy + 6\frac{dy}{dx} + \frac{d}{dx}(9x)$$
Step 2.4:
Determine the derivative of $9x$ with respect to $x$.
Step 2.4.1:
As 9 is a constant, use the constant multiple rule to get $9\frac{dx}{dx}$.
$$2yy + 6y + 9\frac{dx}{dx}$$
Step 2.4.2:
Apply the Power Rule to $x^1$, yielding $1x^{1-1}$.
$$2yy + 6y + 9 \cdot 1$$
Step 2.4.3:
Multiply 9 by 1 to simplify.
$$2yy + 6y + 9$$
Step 3:
The derivative of a constant, 0, is 0.
$$0$$
Step 4:
Combine the derivatives to form the equation.
$$2yy + 6y + 9 = 0$$
Step 5:
Isolate $y$ to solve for $\frac{dy}{dx}$.
Step 5.1:
Subtract 9 from both sides.
$$2yy + 6y = -9$$
Step 5.2:
Factor out $2y$ from the terms on the left side.
Step 5.2.1:
Factor $2y$ from $2yy$.
$$2yy + 6y = -9$$
Step 5.2.2:
Factor $2y$ from $6y$.
$$2yy + 2y \cdot 3 = -9$$
Step 5.2.3:
Extract $2y$ from the sum.
$$2y(y + 3) = -9$$
Step 5.3:
Divide by $2(y + 3)$ and simplify.
Step 5.3.1:
Divide each term by $2(y + 3)$.
$$\frac{2y(y + 3)}{2(y + 3)} = \frac{-9}{2(y + 3)}$$
Step 5.3.2:
Simplify the left side by canceling out common factors.
Step 5.3.2.1:
Cancel the 2.
$$\frac{\cancel{2}y(y + 3)}{\cancel{2}(y + 3)} = \frac{-9}{2(y + 3)}$$
Step 5.3.2.2:
Cancel the $(y + 3)$.
$$\frac{y\cancel{(y + 3)}}{\cancel{(y + 3)}} = \frac{-9}{2(y + 3)}$$
Step 5.3.3:
Simplify the right side.
$$y = -\frac{9}{2(y + 3)}$$
Step 6:
Replace $y$ with $\frac{dy}{dx}$ to find the derivative.
$$\frac{dy}{dx} = -\frac{9}{2(y + 3)}$$
Knowledge Notes:
Sum Rule: The derivative of a sum of functions is the sum of the derivatives of each function.
Chain Rule: Used to differentiate composite functions. If $f(x) = h(g(x))$, then $f'(x) = h'(g(x))g'(x)$.
Power Rule: The derivative of $x^n$ is $nx^{n-1}$.
Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.
Simplification: After differentiation, algebraic simplification may be necessary to isolate the variable of interest, in this case, $\frac{dy}{dx}$.
Implicit Differentiation: When differentiating an equation with respect to $x$ that involves $y$, and $y$ is a function of $x$, the derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.
