Find dy/dx 4x^3+7y^3=11xy

The question asks for the derivative of y with respect to x (dy/dx), which involves implicit differentiation. Implicit differentiation is a technique used when a function is not given in the standard form y=f(x), but rather both x and y are mingled together in an equation like the one provided. The equation in question is a non-linear algebraic equation of x and y, 4x^3 + 7y^3 = 11xy, and you are supposed to apply differentiation rules to both sides of the equation with respect to x, treating y as a function of x. This will involve using the chain rule for differentiating the terms involving y.

$4 x^{3} + 7 y^{3} = 11 x y$

Answer

Expert–verified

Solution:

Step:1

Apply differentiation to both sides of the given equation: $\frac{d}{d x} (4 x^{3} + 7 y^{3}) = \frac{d}{d x} (11 x y)$ .

Step:2

Differentiate the left-hand side term by term.

Step:2.1

Use the Sum Rule to separate the derivatives: $\frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (7 y^{3})$ .

Step:2.2

Find the derivative of $4 x^{3}$ .

Step:2.2.1

Extract the constant $4$ and differentiate $x^{3}$ : $4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (7 y^{3})$ .

Step:2.2.2

Apply the Power Rule to $x^{3}$ : $4 (3 x^{2}) + \frac{d}{d x} (7 y^{3})$ .

Step:2.2.3

Simplify the expression: $12 x^{2} + \frac{d}{d x} (7 y^{3})$ .

Step:2.3

Now, differentiate $7 y^{3}$ .

Step:2.3.1

Extract the constant $7$ and differentiate $y^{3}$ : $12 x^{2} + 7 \frac{d}{d x} (y^{3})$ .

Step:2.3.2

Utilize the Chain Rule for $y^{3}$ .

Step:2.3.2.1

Introduce $u = y$ : $12 x^{2} + 7 (\frac{d}{d u} (u^{3}) \frac{d x}{d y})$ .

Step:2.3.2.2

Apply the Power Rule to $u^{3}$ : $12 x^{2} + 7 (3 u^{2} \frac{d x}{d y})$ .

Step:2.3.2.3

Substitute $u$ back with $y$ : $12 x^{2} + 7 (3 y^{2} \frac{d x}{d y})$ .

Step:2.3.3

Express $\frac{d x}{d y}$ as $y^{'}$ : $12 x^{2} + 7 (3 y^{2} y^{'})$ .

Step:2.3.4

Perform the multiplication: $12 x^{2} + 21 y^{2} y^{'}$ .

Step:3

Differentiate the right-hand side.

Step:3.1

Factor out the constant $11$ : $11 \frac{d}{d x} (x y)$ .

Step:3.2

Apply the Product Rule to $x y$ : $11 (x \frac{d}{d x} (y) + y \frac{d}{d x} (x))$ .

Step:3.3

Replace $\frac{d}{d x} (y)$ with $y^{'}$ : $11 (x y^{'} + y)$ .

Step:3.4

Differentiate $x$ using the Power Rule: $11 (x y^{'} + y \cdot 1)$ .

Step:3.5

Simplify the expression: $11 (x y^{'} + y)$ .

Step:3.6

Distribute the $11$ : $11 x y^{'} + 11 y$ .

Step:4

Combine the differentiated left and right sides: $12 x^{2} + 21 y^{2} y^{'} = 11 x y^{'} + 11 y$ .

Step:5

Isolate $y^{'}$ (dy/dx).

Step:5.1

Subtract $11 x y^{'}$ from both sides: $12 x^{2} + 21 y^{2} y^{'} - 11 x y^{'} = 11 y$ .

Step:5.2

Subtract $12 x^{2}$ from both sides: $21 y^{2} y^{'} - 11 x y^{'} = 11 y - 12 x^{2}$ .

Step:5.3

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

Step:5.3.1

Factor $y^{'}$ from $21 y^{2} y^{'}$ : $y^{'} (21 y^{2}) - 11 x y^{'} = 11 y - 12 x^{2}$ .

Step:5.3.2

Factor $y^{'}$ from $- 11 x y^{'}$ : $y^{'} (21 y^{2} - 11 x) = 11 y - 12 x^{2}$ .

Step:5.3.3

Complete the factoring: $y^{'} (21 y^{2} - 11 x) = 11 y - 12 x^{2}$ .

Step:5.4

Divide by the factored expression to solve for $y^{'}$ .

Step:5.4.1

Divide each term by $21 y^{2} - 11 x$ : $\frac{y^{'} (21 y^{2} - 11 x)}{21 y^{2} - 11 x} = \frac{11 y - 12 x^{2}}{21 y^{2} - 11 x}$ .

Step:5.4.2

Simplify the left side.

Step:5.4.2.1

Cancel the common factors: $\frac{y^{'} (21 y^{2} - 11 x)}{(21 y^{2} - 11 x)} = \frac{11 y - 12 x^{2}}{21 y^{2} - 11 x}$ .

Step:5.4.2.1.2

Simplify to $y^{'}$ : $y^{'} = \frac{11 y - 12 x^{2}}{21 y^{2} - 11 x}$ .

Step:5.4.3

The right side is already simplified.

Step:6

Replace $y^{'}$ with $\frac{d y}{d x}$ : $\frac{d y}{d x} = \frac{11 y - 12 x^{2}}{21 y^{2} - 11 x}$ .

Knowledge Notes:

The process of finding $\frac{d y}{d x}$ for the given implicit function $4 x^{3} + 7 y^{3} = 11 x y$ involves several calculus rules:

Sum Rule: The derivative of a sum is the sum of the derivatives.
Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.
Power Rule: The derivative of $x^{n}$ with respect to $x$ is $n x^{n - 1}$ .
Product Rule: The derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second.
Chain Rule: The derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

In this problem, we differentiate both sides of the equation with respect to $x$ , apply the above rules, and then solve for $\frac{d y}{d x}$ , which represents the derivative of $y$ with respect to $x$ . This process is used in calculus to find the slope of the tangent line to a curve at a given point or to solve related rates problems.