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Section 2.5- Implicit Differentiation

Section 2.5- Implicit Differentiation Thus far, we have differentiated functions in EXPLICIT FORM. y = 4x 3 – x – 4 (the variable y is explicitly written as a function of x ) Explicit- y is by itself as a function of x (on the other side) Example of IMPLICIT FORM:

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Section 2.5- Implicit Differentiation

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  1. Section 2.5- Implicit Differentiation Thus far, we have differentiated functions in EXPLICIT FORM. y = 4x3 – x – 4 (the variable y is explicitly written as a function of x) Explicit- y is by itself as a function of x (on the other side) Example of IMPLICIT FORM: xy = 1 (y is not = something in terms of x -Rewrite this in explicit form and differentiate What about : x2 – 4y4 – 5y = 2 (this is difficult to express in “y = ” form (explicit form) Instead we can use…..…….

  2. …. IMPLICIT DIFFERENTIATION To understand how to find implicitly, you must realize that the differentiation is taking place with respect to x. So, 1) Differentiate x terms involving x alone as usual. 2) But, when you differentiate terms involving y, you must apply the __________ __________, because you are assuming y is defined implicitly as a differentiable function.

  3. Differentiating with respect to X: We will be able to differentiate a function without even knowing what y is in terms of x. Notice our expression for depends not only on x (like explicit functions) but on y, also. 1) 2) 3) 4)

  4. Guidelines for Implicit Differentiation • Differentiate both sides of the equation with respect to x • Collect all terms involving on the left side of the equation and move all other terms to the right side • Factor out of the left side • Solve for by dividing both sides of the equation by the factor on the left hand side that does not contain ( )

  5. Example 1 Find given that y3 + y2 – 5y – x2 = -4

  6. Example 2 Find when sin x + cox y – y2 = 4x3

  7. Implicit vs Explicit Differentiation Example 3 x2 + y2 = 1

  8. Example 4 Find the slope of the tangent line implicitly for: x2 + 4y2 = 4 at the point ( , )

  9. Example 5 Find the slope of the tangent line implicitly for: 3(x2 +y2)2 = 100xy at the point ( , )

  10. Example 6 Find the derivative of sin y = x implicitly.

  11. Example 7 Find the 2nd derivative of the following implicitly: x2 + y2 = 25

  12. Example 8 Find the tangent line to the graph given by: x2(x2 + y2) = y2at the point ( , )

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