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Math Help - Implict differentiation

  1. #1
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    Implict differentiation

    Use implict diffeentiation to find dy/dx

    x^2(cosy)-y^2=e^2x
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  2. #2
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    Quote Originally Posted by Emmeyh15@hotmail.com View Post
    Use implict diffeentiation to find dy/dx

    x^2(cosy)-y^2=e^2x
     \frac{d}{dx} ( x^2(\cos(y) - y^2) = \frac{d}{dx} e^{2x}

    \to  \frac{d}{dx} ( x^2(\cos(y))  - \frac{d}{dx}(y^2) = \frac{d}{dx} e^{2x}

    The first term requies the product rule!

    \to x^2 \frac{d}{dx} (\cos(y)) + \cos(y)\frac{d}{dx}(x^2)  - \frac{d}{dx}(y^2) = \frac{d}{dx} e^{2x}

    The RHS can be written:

    \to x^2 \frac{d}{dx} (\cos(y)) + \cos(y)\frac{d}{dx}(x^2)  - \frac{d}{dx}(y^2) = \frac{d}{dx} (e^{x})^2

    And hence requires the chain rule:

    \to x^2 \frac{d}{dx} (\cos(y)) + \cos(y)\frac{d}{dx}(x^2)  - \frac{d}{dx}(y^2) = 2e^x \times \frac{d}{dx} (2x)

    The first term also requires the chain rule:

    [tex] \to x^2 (\frac{d}{dy} (\cos(y)) \times \frac{d}{dx}(y)) + \cos(y)\frac{d}{dx}(x^2)  - \frac{d}{dx}(y^2) = 2e^{x} \times \frac{d}{dx} (2x)

    Now all you have to do is carry out these relatively simple differentiations, and then change the equation so that you get  \frac{dy}{dx} = .
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  3. #3
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    im confused on how to apply the chain rule to that? i got the poblem right so far but i'm stuck at that point
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  4. #4
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    Quote Originally Posted by Emmeyh15@hotmail.com View Post
    im confused on how to apply the chain rule to that? i got the poblem right so far but i'm stuck at that point
    well the first term on the LHS contains:

     \frac{d}{dx} \cos(y)

    The chain rule says that if you have a function  h(x) = f(g(x)) , then  h'(x) = f'(g(x)) \times g'(x)

    In our case  g(x) = y , and  f(g(x)) = \cos(g(x))

    And hence  g'(x) = \frac{dy}{dx} and  f'(g(x)) = -\sin(g(x))

    Hence
     \frac{d}{dx} \cos(y)  = -\sin(y) \times \frac{dy}{dx}

    In the RHS, we have the same sort of idea.

     g(x) = e^x and  f(g(x)) = (g(x))^2

    Hence:

     g'(x) = e^x , and  f'(g(x)) = 2g(x)

    So the result is :

     frac{d}{dx} (e^x)^2 = e^x \times 2e^x = 2(e^x)^2 = 2e^{2x}
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