Find dy/dx y(x^2+16)=32
The given problem is asking for the derivative of the function with respect to x, symbolized as dy/dx. The function in question has the form y multiplied by a quantity involving x, which in this case is x^2+16, and this product is equal to a constant value, 32. To find dy/dx, you will need to differentiate both sides of the equation with respect to x, applying the appropriate rules for differentiation such as the product rule, the chain rule, and/or the constant multiple rule as necessary.
$y \left(\right. x^{2} + 16 \left.\right) = 32$
Answer
Solution:
Step 1:
Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y(x^2 + 16)) = \frac{d}{dx}(32)$.
Step 2:
Apply the derivative to the left-hand side.
Step 2.1:
Utilize the Product Rule, which is $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = y$ and $v = x^2 + 16$. This gives us $y\frac{d}{dx}(x^2 + 16) + (x^2 + 16)\frac{d}{dx}(y)$.
Step 2.2:
Proceed to differentiate.
Step 2.2.1:
Invoke the Sum Rule to find the derivative of $x^2 + 16$ as $\frac{d}{dx}(x^2) + \frac{d}{dx}(16)$. This results in $y(2x + \frac{d}{dx}(16)) + (x^2 + 16)\frac{d}{dx}(y)$.
Step 2.2.2:
Apply the Power Rule, which tells us that $\frac{d}{dx}(x^n) = nx^{n-1}$ for $n = 2$. This simplifies to $y(2x + \frac{d}{dx}(16)) + (x^2 + 16)\frac{d}{dx}(y)$.
Step 2.2.3:
Recognize that the derivative of a constant is zero, so $\frac{d}{dx}(16) = 0$. This simplifies to $y(2x) + (x^2 + 16)\frac{d}{dx}(y)$.
Step 2.2.4:
Simplify the expression.
Step 2.2.4.1:
Combine $2x$ and $0$ to get $y(2x) + (x^2 + 16)\frac{d}{dx}(y)$.
Step 2.2.4.2:
Rearrange to place the constant $2$ before $y$: $2yx + (x^2 + 16)\frac{d}{dx}(y)$.
Step 2.3:
Express $\frac{d}{dx}(y)$ as $y'$.
Step 2.4:
Simplify further.
Step 2.4.1:
Apply the distributive property to get $2xy + x^2y' + 16y'$.
Step 2.4.2:
Reorder the terms to $2xy + x^2y' + 16y'$.
Step 3:
The derivative of a constant is zero, so $\frac{d}{dx}(32) = 0$.
Step 4:
Combine the derived terms to form the equation $2xy + x^2y' + 16y' = 0$.
Step 5:
Isolate $y'$.
Step 5.1:
Subtract $2xy$ from both sides to get $x^2y' + 16y' = -2xy$.
Step 5.2:
Factor out $y'$ from the left side.
Step 5.2.1:
Extract $y'$ from $x^2y'$ to get $y'(x^2) + 16y' = -2xy$.
Step 5.2.2:
Extract $y'$ from $16y'$ to get $y'(x^2) + y'(16) = -2xy$.
Step 5.2.3:
Factor out $y'$ completely to obtain $y'(x^2 + 16) = -2xy$.
Step 5.3:
Divide through by $x^2 + 16$ and simplify.
Step 5.3.1:
Divide each term to get $\frac{y'(x^2 + 16)}{x^2 + 16} = \frac{-2xy}{x^2 + 16}$.
Step 5.3.2:
Simplify the left side by canceling out the common factors.
Step 5.3.2.1:
Cancel the common factor to get $\frac{y'(\cancel{x^2 + 16})}{\cancel{x^2 + 16}} = \frac{-2xy}{x^2 + 16}$.
Step 5.3.2.1.2:
Reduce $y'$ over $1$ to get $y' = \frac{-2xy}{x^2 + 16}$.
Step 5.3.3:
Simplify the right side by bringing the negative sign to the front of the fraction.
Step 6:
Substitute $y'$ with $\frac{dy}{dx}$ to get the final result: $\frac{dy}{dx} = -\frac{2xy}{x^2 + 16}$.
Knowledge Notes:
Product Rule: When taking the derivative of a product of two functions, $u$ and $v$, the derivative is $u'v + uv'$.
Sum Rule: The derivative of the sum of two functions is the sum of their derivatives.
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.
Distributive Property: In an expression of the form $a(b + c)$, the distributive property allows us to expand this to $ab + ac$.
Factoring: This involves taking a common factor out of terms to simplify expressions or equations.
Simplifying Fractions: When a term in the numerator and the denominator are the same, they can be canceled out.
Negative Sign in Fractions: A negative sign in the numerator or denominator of a fraction can be taken out in front of the fraction.
