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Section 2.5 – Implicit Differentiation
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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).
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Can we take the derivative of these functions?
Implicit Equations Some functions, however, are defined implicitly ( not in the form y = f(x) ) by a relation between x and y such as: 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.
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White Board Challenge Solve for y:
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*Reminder* Technically the Chain Rule can be applied to every derivative:
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Derivatives Involving the Dependent Variable (y)
Find the derivative of each expression 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.
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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/dx on the left side of the equation, and move the other terms to the right side. Factor dy/dx out of the left side Solve for dy/dx
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Example 1 If is a differentiable function of x such that find .
Differentiate both sides. Product AND Constant Multiple Rules Chain Rule Solve for dy/dx
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Example 2 Find if . Differentiate both sides Product Rule
Chain Rule Twice
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Example 3 Find if Find the first derivative by Differentiating both sides. Quotient Rule Chain Rule Remember: Remember: Now Find the Second Derivative
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Example 4 Find the slope of a line tangent to the circle at the point Find the derivative by differentiating both sides. Chain Rule Evaluate the derivative at x=5 and y=4.
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White Board Challenge Find the derivative of:
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Find the derivative by differentiating both sides.
Example 5 If and , find Find the derivative by differentiating both sides. Chain Rule
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Evaluate the derivative with the given information.
Example 5 (continued) If and , find Evaluate the derivative with the given information.
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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
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White Board Challenge Find the second derivative of:
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