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Unformatted text preview: L17 – Implicit Diﬀerentiation Explicit Equations Implicit Equations Assume y is a diﬀerentiable function of x 1) Diﬀerentiate with respect to x on both sides of the equation. dy terms on one side dx dy dx 2) Collect all 3) Factor out 4) Solve for dy dx 1 d x dx d2 x dx d y dx d2 y dx Find dy if x2 + y 2 = 9 . dx a) Implicitly 2 b) Explicitly c) Show that they are the same. 3 Find the horizontal and vertical tangent lines to x2 − 2x + y 2 + 6y = 15 . 4 Find the slope of the tangent line to 2x2 − xy = x + 4 at ( 1, −1 ) . y3 5 Find dy if y = tan(xy ) . dx Two curves are called orthogonal if at each point of intersection, their tangent lines are perpendicular. Two families of curves are orthogonal trajectories if every curve in one family is orthogonal to every curve in the other. 6 Show that the hyperbolas xy = c for c = 0 are orthogonal trajectories to the hyperbolas x2 − y 2 = k for k = 0 . 7 Find d sin−1 (x) dx 8 If f (x) = sin−1 (2x − 1), ﬁnd the derivative, the domain of f (x) and the domain of its derivative. d cos−1 (x) = dx 9 Find the derivative of y = tan−1 (x) . Find the derivative of f (x) = tan−1 ( x − √ 1 + x2 ) . 10 d csc−1 (x) = dx d sec−1 (x) = dx d cot−1 (x) = dx 11 If g (x) = x tan−1 (x), ﬁnd g (x) . 12 ...
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 Fall '08
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 Calculus, Equations, Derivative, Slope, Mathematical analysis

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