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please find `dy/dx` using implicit differentiation, and please explain the use of the...
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(Level 1) Associate Educator, Expert
Let D(f(x)) = f'(x) be the derivative of f(x).
We take the derivative of both sides of the equation:
`D(sin(x^4y^4)) = D(x)`
`cos(x^4y^4) D(x^4y^4) = 1`
`cos(x^4y^4) [4x^4y^3 (dy)/(dx) + 4y^4x^3] = 1`
`(dy)/(dx) = [(1/(cos(x^4y^4))) - 4y^4x^3]/(4x^4y^3)`
Here, we used the chain rule in evaluating the derivative of `sin(x^4y^4)`.
We can think of sin(x) as the outer function, with inner function `x^4y^4` . The chain rule states that the derivative of f(g(x)) with respect to x is the derivative of f(x) multiplied by the derivative of g(x).
In this case, we first got the derivative of the outer function:
`D(sin(x^4y^4)) = cos(x^4y^4)`
and multipled it with the derivative of the inner function (which we got by applying the product rule, and noting that we are getting the implicit derivative.
Posted by mvcdc on July 1, 2013 at 5:40 PM (Answer #1)
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