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    Implicit differentiation

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    Find dy/dx by implicit differentiation, please see the remaining questions in the attachment.

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    https://brainmass.com/physics/right-hand-rule/implicit-differentiation-465137

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    To do implicit differentiation, you differentiate all terms in the equation. For the x-terms, you just do regular differentiation but for the y-terms you need to apply the chain rule because y = f(x), y is a function of x.

    1. Find dy/dx by implicit differentiation.
    (11x + 2y)^1/3 = x^2

    Solution:

    Differentiating both sides of the equation we get:

    For the left hand side, you apply the chain rule and for the right hand side, you apply the power rule.

    1/3 (11x + 2y)^-2/3 (11 + 2y') = 2x

    Now solve for y':

    11 + 2 y' = 6x (11x + 2y)^2/3

    2 y' = 6x (11x + 2y)^2/3 - 11

    Now divide both sides by 2:

    y' = 3x(11x + 2y)^2/3 - 11/2 is the solution

    2. Find dy/dx by implicit differentiation.
    (x + y^2)^10 = 7x^2 + 2

    Solution:

    Differentiating both sides of the equation we get:

    For the left hand side, you apply the chain rule and for the right hand side, you apply the power rule.
    10(x + y^2)^9 (1 + 2y y') = 14x

    Now solve for y':
    1 + 2y y' = 14x/[10(x + y^2)^9]

    1 + 2y y' = 7x/[5(x + y^2)^9]

    2y y' = 7x/[5(x + y^2)^9] - 1

    y' = 7x/[10y(x + y^2)^9] - 1/2y is the solution

    3. Find an equation of the tangent ...

    Solution Summary

    The expert finds the dy/dx by implicit differentiation.

    $2.49

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