Use implicit differentiation to find d y / d x in Exercises 1-14 x^3=(2 x-y)/(1-x^2)

Name: Problem 8, Chapter 3: Calculus
Uploaded: 2020-08-01T10:36:55-08:00
Duration: 4 min 13 s
Channel: Darshan Maheshwari
Description: Use implicit differentiation to find d y / d x in Exercises 1-14 x^3=(2 x-y)/(1-x^2)

step 1 Let's talk about problem 8. This one says that x cubed is equal to 2x minus y over 1 minus x squ

Use implicit differentiation to find d y / d x in Exercises...

Key Concepts

Implicit Differentiation

A technique used to differentiate equations in which y is not isolated as a function of x. Instead of explicitly solving for y, both sides of the equation are differentiated with respect to x while treating y as a function of x, which allows the derivative dy/dx to be computed even when the equation is given in an implicit form.

Chain Rule

A fundamental rule in calculus that is used to differentiate composite functions. In the context of implicit differentiation, the chain rule is applied when differentiating terms that involve y, since y itself is a function of x. This process involves differentiating the outer function and then multiplying by the derivative of the inner function.

Quotient Rule

A method for differentiating expressions formed by the ratio of two functions. When an implicit equation contains a fraction, the quotient rule aids in correctly finding the derivative by taking the derivative of the numerator and the denominator in a specific formula, ensuring that the differentiation is carried out systematically.

Algebraic Manipulation

After applying the necessary differentiation rules, the resulting equation typically contains terms involving dy/dx. Algebraic manipulation involves rearranging and simplifying this equation—combining like terms, factoring, or isolating dy/dx—so that the derivative can be expressed explicitly.

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