Implicit Differentiation?

In summary, implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly written in terms of one variable. It is used when the dependent variable cannot be easily isolated and the function is written in implicit form. This differs from explicit differentiation, which is used for functions written explicitly in terms of one variable. The steps for performing implicit differentiation involve identifying the variables, differentiating with respect to the independent variable, and solving for the derivative. Real-life applications of implicit differentiation can be found in fields such as physics, engineering, and economics, where it is used to analyze and model complex relationships between variables.
  • #1
JessicaJ283782
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Homework Statement



Find the derivative of:

x+xy=y^2


Homework Equations




So I know you have to differentiate it, and it would be:

1+xyy'=2yy'

The Attempt at a Solution




Moving the terms with y' to one side:
1+xyy'-2yy'=0

xyy'-2yy'=-1

Factoring out y'

y'(xy-2y)=-1

y'=-1/(xy-2y)

Did I do that correctly?
 
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  • #2
The derivative of xy is not xyy' but it is y + xy'
 
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  • #3
For the xy part, you should use the product rule, f'g+g'f
 

1. What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly written in terms of one variable. In other words, it is used to find the rate of change of a function when the dependent variable cannot be easily isolated.

2. When is implicit differentiation used?

Implicit differentiation is used when a function is written in implicit form, meaning that the dependent variable is not explicitly expressed in terms of the independent variable. This commonly occurs in equations involving multiple variables or when the function is defined implicitly.

3. How does implicit differentiation differ from explicit differentiation?

Explicit differentiation is used to find the derivative of a function that is written explicitly in terms of one variable. Implicit differentiation is used when the function is not explicitly written in terms of one variable. Additionally, implicit differentiation often involves using the chain rule and product rule, whereas explicit differentiation typically only involves the power rule and basic derivatives.

4. What are the steps for performing implicit differentiation?

The steps for performing implicit differentiation are as follows:

  1. Identify the dependent and independent variables in the function.
  2. Differentiate both sides of the equation with respect to the independent variable.
  3. Use the chain rule and product rule if necessary.
  4. Solve for the derivative of the dependent variable.

5. What are some real-life applications of implicit differentiation?

Implicit differentiation is used in many fields, such as physics, engineering, and economics, to analyze and model complex relationships between variables. For example, it can be used to find the rate of change of a variable in a physical system or to optimize production functions in economics. It is also commonly used in multivariable calculus to solve optimization problems.

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