Find dy/dx y(x^2+36)=72
This problem is asking for the differentiation of a given function with respect to x. The function in question is an implicit function where y is multiplied by a binomial expression consisting of x^2 and a constant (36). The result of this multiplication is set equal to another constant (72). To find dy/dx (the derivative of y with respect to x), one must use implicit differentiation, which involves differentiating both sides of the equation while applying the chain rule, product rule, and taking into consideration the derivative of y with respect to x whenever y is differentiated.
$y \left(\right. x^{2} + 36 \left.\right) = 72$
Answer
Solution:
Step 1
Apply differentiation to both sides of the given equation: $y(x^2 + 36) = 72$.
$$\frac{d}{dx} [y(x^2 + 36)] = \frac{d}{dx} [72]$$
Step 2
Differentiate the left-hand side using the product rule.
Step 2.1
Invoke the product rule: $\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = y$ and $v = x^2 + 36$.
$$y \frac{d}{dx}[x^2 + 36] + (x^2 + 36) \frac{d}{dx}[y]$$
Step 2.2
Perform the differentiation.
Step 2.2.1
Apply the sum rule to the derivative of $x^2 + 36$.
$$y(\frac{d}{dx}[x^2] + \frac{d}{dx}[36]) + (x^2 + 36) \frac{d}{dx}[y]$$
Step 2.2.2
Differentiate $x^2$ using the power rule: $\frac{d}{dx}[x^n] = nx^{n-1}$.
$$y(2x + \frac{d}{dx}[36]) + (x^2 + 36) \frac{d}{dx}[y]$$
Step 2.2.3
Recognize that the derivative of a constant is zero.
$$y(2x + 0) + (x^2 + 36) \frac{d}{dx}[y]$$
Step 2.2.4
Simplify the expression.
Step 2.2.4.1
Combine $2x$ and $0$.
$$2yx + (x^2 + 36) \frac{d}{dx}[y]$$
Step 2.2.4.2
Rearrange the terms for clarity.
$$2yx + (x^2 + 36) \frac{d}{dx}[y]$$
Step 2.3
Express $\frac{d}{dx}[y]$ as $y'$.
$$2yx + (x^2 + 36)y'$$
Step 2.4
Simplify the expression further.
Step 2.4.1
Distribute $y$ across the sum.
$$2xy + x^2y' + 36y'$$
Step 2.4.2
Reorder the terms for clarity.
$$2xy + x^2y' + 36y'$$
Step 3
The derivative of a constant is zero.
$$\frac{d}{dx}[72] = 0$$
Step 4
Combine the differentiated left-hand side with the right-hand side.
$$2xy + x^2y' + 36y' = 0$$
Step 5
Isolate $y'$.
Step 5.1
Subtract $2xy$ from both sides.
$$x^2y' + 36y' = -2xy$$
Step 5.2
Factor out $y'$.
Step 5.2.1
Extract $y'$ from $x^2y'$.
$$y'(x^2 + 36) = -2xy$$
Step 5.2.2
Factor $y'$ completely.
$$y'(x^2 + 36) = -2xy$$
Step 5.2.3
Write the factored form.
$$y'(x^2 + 36) = -2xy$$
Step 5.3
Divide by $(x^2 + 36)$ to solve for $y'$.
Step 5.3.1
Divide both sides by $(x^2 + 36)$.
$$\frac{y'(x^2 + 36)}{x^2 + 36} = \frac{-2xy}{x^2 + 36}$$
Step 5.3.2
Simplify the left side by canceling out common factors.
Step 5.3.2.1
Cancel $(x^2 + 36)$.
$$\frac{y' \cancel{(x^2 + 36)}}{\cancel{(x^2 + 36)}} = \frac{-2xy}{x^2 + 36}$$
Step 5.3.2.2
Simplify to $y'$.
$$y' = \frac{-2xy}{x^2 + 36}$$
Step 5.3.3
Simplify the right side.
Step 5.3.3.1
Place the negative sign in front of the fraction.
$$y' = -\frac{2xy}{x^2 + 36}$$
Step 6
Replace $y'$ with $\frac{dy}{dx}$.
$$\frac{dy}{dx} = -\frac{2xy}{x^2 + 36}$$
Knowledge Notes:
The problem involves finding the derivative of a function that is given in an implicit form, $y(x^2 + 36) = 72$. The process of solving this problem requires knowledge of several calculus concepts:
Derivative: The derivative of a function measures how the function value changes as its input changes. The notation $\frac{d}{dx}$ is used to denote the derivative with respect to $x$.
Product Rule: When differentiating a product of two functions, $u(x)v(x)$, the derivative is given by $u'v + uv'$, where $u'$ and $v'$ are the derivatives of $u$ and $v$ respectively.
Sum Rule: The derivative of a sum of functions is the sum of the derivatives of those functions.
Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Constant Rule: The derivative of a constant is zero.
Implicit Differentiation: When a function is not given explicitly as $y=f(x)$, but rather in a form that involves both $x$ and $y$, we use implicit differentiation to find $\frac{dy}{dx}$.
Factoring: Factoring is the process of expressing an expression as the product of its factors. It is used to simplify expressions and solve equations.
Simplification: The process of reducing an expression to its simplest form by performing operations like addition, subtraction, multiplication, division, and canceling common factors.
By applying these concepts, we can differentiate the given implicit function and solve for $\frac{dy}{dx}$.
