Find dy/dx y(x^2+16)=32

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 $\frac{d}{dx}(x^n)=n{x}^{n-1}$ 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 $y^{\prime }$.

Step 2.4:

Simplify further.

Step 2.4.1:

Apply the distributive property to get $2xy+x^2{y}^{\prime }+16y^{\prime }$.

Step 2.4.2:

Reorder the terms to $2xy+x^2{y}^{\prime }+16y^{\prime }$.

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 $2xy+x^2{y}^{\prime }+16y^{\prime }=0$.

Step 5:

Isolate $y^{\prime }$.

Step 5.1:

Subtract $2xy$ from both sides to get $x^2{y}^{\prime }+16y^{\prime }=-2xy$.

Step 5.2:

Factor out $y^{\prime }$ from the left side.

Step 5.2.1:

Extract $y^{\prime }$ from $x^2{y}^{\prime }$ to get $y^{\prime }(x^2)+16y^{\prime }=-2xy$.

Step 5.2.2:

Extract $y^{\prime }$ from $16y^{\prime }$ to get $y^{\prime }(x^2)+y^{\prime }(16)=-2xy$.

Step 5.2.3:

Factor out $y^{\prime }$ completely to obtain $y^{\prime }(x^2+16)=-2xy$.

Step 5.3:

Divide through by $x^2+16$ and simplify.

Step 5.3.1:

Divide each term to get $\frac{y^{\prime }(x^2+16)}{x^2+16}=\frac{-2xy}{x^2+16}$.

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 $\frac{y^{\prime }(\cancel{x^2+16})}{\cancel{x^2+16}}=\frac{-2xy}{x^2+16}$.

Step 5.3.2.1.2:

Reduce $y^{\prime }$ over $1$ to get $y^{\prime }=\frac{-2xy}{x^2+16}$.

Step 5.3.3:

Simplify the right side by bringing the negative sign to the front of the fraction.

Step 6:

Substitute $y^{\prime }$ with $\frac{dy}{dx}$ to get the final result: $\frac{dy}{dx}=-\frac{2xy}{x^2+16}$.

Knowledge Notes:

1. Product Rule: When taking the derivative of a product of two functions, $u$ and $v$, the derivative is $u^{\prime }v+u{v}^{\prime }$.

2. Sum Rule: The derivative of the sum of two functions is the sum of their derivatives.

3. Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $n{x}^{n-1}$.

4. Constant Rule: The derivative of a constant is zero.

5. Distributive Property: In an expression of the form $a(b+c)$, the distributive property allows us to expand this to $ab+ac$.

6. Factoring: This involves taking a common factor out of terms to simplify expressions or equations.

7. Simplifying Fractions: When a term in the numerator and the denominator are the same, they can be canceled out.

8. Negative Sign in Fractions: A negative sign in the numerator or denominator of a fraction can be taken out in front of the fraction.