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Implicit Differentiation

Implicit Differentiation. Objective. To find derivatives implicitly. Implicit Differentiation. Fails the vertical line test!. Implicit Differentiation. Let’s say we had a circle. Because once I had it I could find the slope of any point, even if it wasn’t obvious from the picture!.

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Implicit Differentiation

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  1. Implicit Differentiation

  2. Objective • To find derivatives implicitly.

  3. Implicit Differentiation Fails the vertical line test!

  4. Implicit Differentiation Let’s say we had a circle

  5. Because once I had it I could find the slope of any point, even if it wasn’t obvious from the picture! I can see the slope at (0, -1) is 0, and I know that a derivative equation gives slope. But how can I find the derivative of that equation?

  6. Implicit Differentiation

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  9. For some equations we could do it the old fashioned way, by isolating y.

  10. Implicit Differentiation • So far in calculus, we have worked only with equations in explicit form. • An equation is in explicit form when one variable is directly equal to an expression made up of the other variable.

  11. Implicit Differentiation • Sometimes equations are not in explicit form, but in a more complicated form in which it is difficult or impossible to express one variable in terms of the other. Such equations are in implicit form.

  12. Implicit Differentiation • Implicit differentiation uses the chain rule in a creative way to find the derivative of equations in implicit form.

  13. Implicit Differentiation

  14. Implicit Differentiation

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  20. Implicit Differentiation Find the equation of the tangent line to the graph of:

  21. Implicit Differentiation

  22. Conclusion • An equation is in explicit form when one variable is directly equal to an expression made up of the other variable. • An equation is in implicit form when neither variable is isolated on one side of the equal sign. • Find the derivative of a relation by differentiating each side of its equation implicitly and solving for the derivative as an unknown.

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