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lecture17

# lecture17 - Functions Find d dx(sin ± 1 x What is d dx(cos...

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Lecture 17: Implicit Di±erentiation (Sec. 3.5) Explicit Functions ex. Find dy dx if y ± x = x 2 sin x . Implicit Functions ex. Consider the equation x 2 y + 3 x = y 4 . If y is a di±erentiable function of x , can we ²nd dy dx ?

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To Di±erentiate Implicitly: Assume y is a di±erentiable function of x . 1. Di±erentiate both sides of the equation with re- spect to x . 2. Collect all terms involving dy dx on one side of the equation. 3. Rewrite by factoring out dy dx . 4. Solve for dy dx . NOTE: d dx ( x ) = and d dx ( x 2 ) = Now suppose that y is a function of x . d dy ( y 2 )= What is d dx ( y 2 )?
ex. Find dy dx if y = tan( xy ).

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ex. Find the slope of the tangent line to x 2 + y 2 = 9 at the point (2 ; ± p 5) a) Explicitly 6 - ? ±
Find the slope of the tangent line to x 2 + y 2 = 9 at the point (2 ; ± p 5) b) Implicitly c) Show that the two forms of dy dx are equivalent.

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ex. Find the slope of the tangent line to 2 x 2 ± xy = x y 3 + 4 at the point (1 ; ± 1).
ex. Find the horizontal and vertical tangent lines of the curve x 2 ± 2 x + y 2 + 6 y = 15. 6 - ? ±

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Derivatives of Inverse Trigonometric

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Unformatted text preview: Functions Find d dx (sin ± 1 x ). What is d dx (cos ± 1 x )? ex. Let f ( x ) = sin ± 1 (2 x ± 1). Find the following: 1) domain of f ( x ) 2) f ( x ) = 3) domain of f ( x ) Find the derivative of y = tan ± 1 x . ex. Find f ( x ) if f ( x ) = tan ± 1 ( e 2 x ). ex. If g ( x ) = x tan ± 1 ( x ), ±nd the equation of the tangent line to g ( x ) at x = 1. d dx (csc ± 1 x ) = d dx (sec ± 1 x ) = d dx (cot ± 1 x ) = ex. Two curves are called orthogonal if at each point of intersection their tangent lines are perpen-dicular. Two familie of curves are orthogonal tra-jectories if every curve in one family is orthogonal to every curve in the other. Show that the hyperbolas xy = c for c 6 = 0 are orthogonal trajectories to the hyperbolas x 2 ± y 2 = k for k 6 = 0....
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lecture17 - Functions Find d dx(sin ± 1 x What is d dx(cos...

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