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Implicit differentiation

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

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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.

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