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3.6 Implicit Differentiation

Discover the power of implicit differentiation & rational exponents, with helpful examples & explanations. Unravel the mystery behind fractional exponents and implicit differentiation. Learn to handle non-integer exponents effectively with the generalized power rule.

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3.6 Implicit Differentiation

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  1. 3.6 Implicit Differentiation And Rational Exponents

  2. Rational Exponents are easy! These problems will have exponents that aren’t integers; they will most likely be fractions. Hmmm… how could an exponent be a fraction?? That’s right! A radical!! It doesn’t matter. You will still use the generalized power rule:

  3. Examples • Find if • Find if Functions do NOT need to be rationalized; only numbers need to be rationalized.

  4. Implicit Differentiation This process is used when y cannot be isolated. We will look at a problem where y CAN be isolated to understand the idea, but in many problems y will be mixed in as a product or a quotient. We will be treating y as a differentiable functions.

  5. Let’s try this one The derivative of y in terms of x has to have that extra symbol because “y” is not “x” Same thing right?

  6. What does this mean? WHENEVER you have to take a derivative of y, tack on . No kidding.  Then, isolate . There might be y in your answer; it also might be easy to sub back in y. Lets see an example… or two….

  7. Examples 3. Find if • Find if • Find if (hint: continue #4) Leave y in the answer.

  8. Examples 6. Find the slope at (-1, 1) for

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