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### Section 3.10 : Implicit Differentiation

6. Find $$y'$$ by implicit differentiation for $${{\bf{e}}^x} - \sin \left( y \right) = x$$.

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Hint : Don’t forget that $$y$$ is really $$y\left( x \right)$$ and so we’ll need to use the Chain Rule when taking the derivative of terms involving $$y$$!
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First, we just need to take the derivative of everything with respect to $$x$$ and we’ll need to recall that $$y$$ is really $$y\left( x \right)$$ and so we’ll need to use the Chain Rule when taking the derivative of terms involving $$y$$.

Differentiating with respect to $$x$$ gives,

${{\bf{e}}^x} - \cos \left( y \right)y' = 1$

Don’t forget the $$y'$$ on the cosine after differentiating. Again, $$y$$ is really $$y\left( x \right)$$ and so when differentiating $$\sin \left( y \right)$$ we really differentiating $$\sin \left[ {y\left( x \right)} \right]$$ and so we are differentiating using the Chain Rule!

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Finally, all we need to do is solve this for $$y'$$.

$\require{bbox} \bbox[2pt,border:1px solid black]{{y' = \frac{{1 - {{\bf{e}}^x}}}{{ - \cos \left( y \right)}} = \left( {{{\bf{e}}^x} - 1} \right)\sec \left( y \right)}}$