Find dy/dx y(x^2+81)=162
This problem is related to differential calculus, specifically, it involves finding the first derivative of a given function with respect to the variable x. The function is provided implicitly rather than explicitly, meaning that y is not isolated on one side of the equation. The equation y(x^2 + 81) = 162 shows that y is a function of x, wherein y is multiplied by a quadratic expression in x. To find dy/dx, which represents the rate of change of y with respect to x, you will need to differentiate both sides of the given equation with respect to x, applying implicit differentiation. This process will likely involve the product rule of differentiation since y, a function of x, is being multiplied by another expression involving x.
$y \left(\right. x^{2} + 81 \left.\right) = 162$
Answer
Solution:
Step 1:
Apply the derivative operator $\frac{d}{dx}$ to both sides of the equation $y(x^2 + 81) = 162$.
Step 2:
Use the product rule for differentiation on the left-hand side, which states that $\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = y$ and $v = x^2 + 81$.
Step 3:
Compute the derivatives separately.
Step 3.1:
The derivative of $x^2 + 81$ is found using the sum rule: $\frac{d}{dx}[x^2] + \frac{d}{dx}[81]$.
Step 3.2:
Apply the power rule to $x^2$, which gives $2x$, and note that the derivative of a constant is zero, so the derivative of $81$ is $0$.
Step 3.3:
Combine the results to get $2xy + (x^2 + 81)\frac{dy}{dx}$.
Step 4:
Differentiate the right-hand side, which is a constant, to get $0$.
Step 5:
Set the differentiated left-hand side equal to the differentiated right-hand side: $2xy + (x^2 + 81)\frac{dy}{dx} = 0$.
Step 6:
Solve for $\frac{dy}{dx}$.
Step 6.1:
Isolate the term containing $\frac{dy}{dx}$.
Step 6.2:
Factor out $y$ from the terms without $\frac{dy}{dx}$.
Step 6.3:
Divide both sides by $x^2 + 81$ to solve for $\frac{dy}{dx}$.
Step 6.4:
Simplify to find $\frac{dy}{dx} = -\frac{2xy}{x^2 + 81}$.
Knowledge Notes:
Product Rule: When differentiating a product of two functions, the derivative is given by $d(uv)/dx = u(dv/dx) + v(du/dx)$.
Sum Rule: The derivative of a sum of functions is the sum of their derivatives, expressed as $d(u + v)/dx = du/dx + dv/dx$.
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. This is because constants do not change, and differentiation measures the rate of change.
Solving for Derivatives: After applying the rules of differentiation, algebraic manipulation is often required to isolate the derivative term and solve for it.
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