How to Find dy/dx using Implicit Differentiation Given y^2 - y*e^x = 12 (Example with Exponentials) | Summary and Q&A

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November 1, 2020
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How to Find dy/dx using Implicit Differentiation Given y^2 - y*e^x = 12 (Example with Exponentials)

TL;DR

Use implicit differentiation to find dy/dx, the derivative of y with respect to x.

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Key Insights

  • 🍗 Implicit differentiation is useful when trying to find derivatives in equations where y is not explicitly defined in terms of x.
  • 😀 The chain rule is crucial in implicit differentiation to account for the derivative of y with respect to x.
  • ❣️ The product rule is applied in cases where functions involving both y and x need to be differentiated.
  • 👻 Factoring out dy/dx simplifies the equation and allows for the isolation of the derivative.
  • 🐞 The final expression for dy/dx involves both y and e^x terms in the numerator and a combination of y and e^x terms in the denominator.
  • ❓ Implicit differentiation can be challenging and requires practice to master.
  • 📏 Understanding how to use the chain rule and product rule is key to successfully perform implicit differentiation.

Transcript

hi everyone in this problem we're being asked to use implicit differentiation to find d y d x so here we can assume that y is a function of x so we start a problem like this by taking the derivative of both sides with respect to x you can just start by taking the derivative or you can actually write down that you're going to take the derivative i'm... Read More

Questions & Answers

Q: What is implicit differentiation used for?

Implicit differentiation is used to find the derivative of a function where y is a function of x, especially when it is difficult or impossible to solve for y explicitly.

Q: How is implicit differentiation different from regular differentiation?

Implicit differentiation treats y as a function of x, so the chain rule is applied to account for the derivative of y. Regular differentiation assumes y is explicitly defined in terms of x.

Q: Why is it necessary to use the chain rule in implicit differentiation?

The chain rule is necessary because y is treated as a function of x. When differentiating a term with y, the chain rule accounts for the derivative of y with respect to x.

Q: How does the product rule apply in this implicit differentiation problem?

The product rule is used when differentiating the term involving y and e^x. It involves taking the derivative of y and multiplying it by e^x, then adding the product of y and the derivative of e^x.

Summary & Key Takeaways

  • Implicit differentiation is used to find the derivative of a function where y is a function of x.

  • The process involves taking the derivative of both sides of the equation and applying the chain rule.

  • The final result is dy/dx = ye^x / (2y - e^x).

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