Implicit Differentiation. Objective To find derivatives implicitly. To find derivatives implicitly.

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Presentation transcript:

Implicit Differentiation

Objective To find derivatives implicitly. To find derivatives implicitly.

Implicit Differentiation Fails the vertical line test!

Implicit Differentiation Let’s say we had a circle

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? Because once I had it I could find the slope of any point, even if it wasn’t obvious from the picture!

Implicit Differentiation

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

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

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. 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.

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

Implicit Differentiation

Find the equation of the tangent line to the graph of:

Implicit Differentiation

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 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. 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. Find the derivative of a relation by differentiating each side of its equation implicitly and solving for the derivative as an unknown.