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Section 2.5 – Implicit Differentiation

Section 2.5 – Implicit Differentiation. Explicit Equations. The functions that we have differentiated and handled so far can be described by expressing one variable explicitly in terms of another variable. For example: Or, in general, y = f ( x ) . Implicit Equations.

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Section 2.5 – Implicit Differentiation

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  1. Section 2.5 – Implicit Differentiation

  2. Explicit Equations The functions that we have differentiated and handled so far can be described by expressing one variable explicitly in terms of another variable. For example: Or, in general, y = f(x).

  3. Implicit Equations Can we take the derivative of these functions? It is possible to solve some Implicit Equations for y, then differentiate: Yet, it is difficult to rewrite most Implicit Equations explicitly. Thus, we must be introduced to a new technique to differentiate these implicit functions. Some functions, however, are defined implicitly ( not in the form y = f(x) ) by a relation between x and y such as:

  4. White Board Challenge Solve for y:

  5. *Reminder* Technically the Chain Rule can be applied to every derivative:

  6. Derivatives Involving the Dependent Variable (y) The Chain Rule is Required. a. b. The derivative of y with respect to x is… the derivative of y. This is another way to write y prime. Find the derivative of each expression

  7. Instructions for Implicit Differentiation If y is an equation defined implicitly as a differentiable function of x, to find the derivative: • Differentiate both sides of the equation with respect to x. (Remember that y is really a function of x for part of the curve and use the Chain Rule when differentiating terms containing y) • Collect all terms involving dy/dxon the left side of the equation, and move the other terms to the right side. • Factor dy/dxout of the left side • Solve for dy/dx

  8. Example 1 Differentiate both sides. Product AND Constant Multiple Rules Chain Rule Solve for dy/dx If is a differentiable function of x such that find .

  9. Example 2 Differentiate both sides Product Rule Chain Rule Twice Find if .

  10. Example 3 Find the first derivative by Differentiating both sides. Quotient Rule Chain Rule Remember: Remember: Find if . Now Find the Second Derivative

  11. Example 4 Find the derivative by differentiating both sides. Chain Rule Evaluate the derivative at x=5 and y=4. Find the slope of a line tangent to the circle at the point .

  12. White Board Challenge Find the derivative of:

  13. Example 5 Find the derivative by differentiating both sides. Chain Rule If and , find .

  14. Example 5 (continued) Evaluate the derivative with the given information. If and , find .

  15. Example 6 Find an equation of the tangent to the circle at the point . Now evaluate the derivative at x=3 and y=4. Find the derivative by differentiating both sides. Chain Rule Use the Point-Slope Formula to find the equation of the tangent line

  16. White Board Challenge Find the second derivative of:

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