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.
Take the derivative of both sides of the equation with respect to
Differentiate each term on the left side separately.
Apply the Sum Rule in differentiation:
Differentiate
The constant
Simplify to get
Differentiate
The constant
Recognize that
Simplify to
Differentiate
The constant
Combine all the differentiated terms.
Combine like terms to simplify the expression:
Differentiate the right side of the equation, which is a constant, to get
Set the left side equal to the right side to form the equation:
Solve for
Isolate terms involving
Move
Combine
Factor out
Factor
Divide both sides by
Divide and simplify:
Sum Rule: The derivative of a sum of functions is the sum of the derivatives.
Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Power Rule: The derivative of
Product Rule: The derivative of a product of two functions
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.
Implicit Differentiation: When a function is not given explicitly as
Derivative of a Constant: The derivative of a constant is zero.
Derivative of