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    Implicit differentiation, equation of the tangent line

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    1) Find f'(x) and f'(c): f(x) = x^2 - 4 / x - 3, c =1
    2) Find dy/dx by implicit differentiation: sinx + 2cos2y = 1

    3)
    A) Use implicit differentiation to find an equation of the tangent line to the ellipse:
    x^2/2 + y^2/8 = 1 at (1,2)

    B) Show that the equation of the tangent line to the ellipse 2^2/a^2 + y^2/b^2 = 1 at (x_0,y_0) is
    x_0 x/ a^2 + y_0 y/b2^2 = 1

    © BrainMass Inc. brainmass.com December 24, 2021, 10:37 pm ad1c9bdddf
    https://brainmass.com/math/derivatives/implicit-differentiation-equation-tangent-line-493852

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    SOLUTION This solution is FREE courtesy of BrainMass!

    6. Find and :
    Applying the quotient rule and chain rule

    Then substitute 1 for x to find out f'(x=1)

    9. Find by implicit differentiation:
    Applying chain rule, and differentiating on both sides of equation:

    So

    10. (a) Use implicit differentiation to find an equation of the tangent line to the ellipse .
    Based the definition, the slope of the tangent line is
    First find the general formula of the slope of the tangent line:
    Differentiate with respect to x,

    Then

    That is, the slope of the tangent line is
    At point (1, 2), the slope of the tangent line is
    Now we just need to find the line equation with slope and passing through point (1, 2). Based on the definition of slope, we have

    That is, the tangent line to the ellipse at point (1, 2) is

    (b) Show that the equation of the tangent line to the ellipse at is .
    First find the slope of the tangent line by differentiating both sides of the ellipse equation:

    So the slope of the tangent line at point is

    Then the line equation with the above specific slope and passing through point is:

    Dividing from both sides of equation:

    Since point is on the ellipse, it then satisfies the equation .
    Hence, the tangent line equation can be written as:

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 10:37 pm ad1c9bdddf>
    https://brainmass.com/math/derivatives/implicit-differentiation-equation-tangent-line-493852

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