Problem

Find dy/dx 3x^2-2xy+3y=1

The question provided is asking for the derivative of the equation 3x^2 - 2xy + 3y = 1 with respect to x, commonly denoted as dy/dx. In calculus, finding dy/dx of an equation involving both x and y variables typically involves differentiating implicitly, because y is considered a function of x (y = f(x)) even though it is not solved for explicitly. The process will require the use of implicit differentiation rules, which take the derivatives of both sides of the equation with respect to x while applying the chain rule to the terms involving y since y is a function of x.

3x22xy+3y=1

Answer

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Solution:

Step:1

Take the derivative of both sides of the equation with respect to x: ddx(3x22xy+3y)=ddx(1).

Step:2

Differentiate each term on the left side separately.

Step:2.1

Apply the Sum Rule in differentiation: ddx(3x2)+ddx(2xy)+ddx(3y).

Step:2.2

Differentiate 3x2 with respect to x.

Step:2.2.1

The constant 3 remains, and the derivative of x2 is 2x, resulting in 3(2x).

Step:2.2.2

Simplify to get 6x.

Step:2.3

Differentiate 2xy with respect to x.

Step:2.3.1

The constant 2 remains, and apply the Product Rule to xy: 2(xddx(y)+yddx(x)).

Step:2.3.2

Recognize that ddx(y) is dydx and ddx(x) is 1.

Step:2.3.3

Simplify to 2(xy+y).

Step:2.4

Differentiate 3y with respect to x.

Step:2.4.1

The constant 3 remains, and the derivative of y is dydx, resulting in 3dydx.

Step:2.5

Combine all the differentiated terms.

Step:2.5.1

Combine like terms to simplify the expression: 6x2(xy+y)+3y.

Step:3

Differentiate the right side of the equation, which is a constant, to get 0.

Step:4

Set the left side equal to the right side to form the equation: 6x2(xy+y)+3y=0.

Step:5

Solve for dydx.

Step:5.1

Isolate terms involving dydx on one side and the rest on the other side.

Step:5.1.1

Move 6x to the right side: 2xy2y+3y=6x.

Step:5.1.2

Combine 2y and 3y: 2xy+y=6x+2y.

Step:5.2

Factor out dydx from the left side.

Step:5.2.1

Factor dydx out of 2xy and y: dydx(2x+1)=6x+2y.

Step:5.3

Divide both sides by 2x+1 to solve for dydx.

Step:5.3.1

Divide and simplify: dydx=6x+2y2x+1.

Knowledge Notes:

  1. Sum Rule: The derivative of a sum of functions is the sum of the derivatives.

  2. Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

  3. Power Rule: The derivative of xn with respect to x is nxn1.

  4. Product Rule: The derivative of a product of two functions f(x)g(x) is f(x)g(x)+f(x)g(x).

  5. 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.

  6. Implicit Differentiation: When a function is not given explicitly as y=f(x), but implicitly as a relation between x and y, we differentiate both sides of the equation with respect to x and solve for dydx.

  7. Derivative of a Constant: The derivative of a constant is zero.

  8. Derivative of y with respect to x: When differentiating y with respect to x, we denote it as dydx, which represents the rate of change of y with respect to x.

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