vasorasy8

2022-06-30

If $x{e}^{9y}+{y}^{4}\mathrm{sin}\left(4x\right)={e}^{8x}$ implicitly defines y as a function of x then what is $\frac{dy}{dx}$?

Keely Fernandez

Expert

Implicitl differentiation means that we need to treat y as a function of x and apply Chain Rule. You also forgot to apply Product Rule to the first term. The derivative of the first term is:
$\begin{array}{rl}\frac{d}{dx}\left[x{e}^{9y}\right]& =x\left[\frac{d}{dx}{e}^{9y}\right]+\left[\frac{d}{dx}x\right]{e}^{9y}\\ & =x{e}^{9y}\left[\frac{d}{dx}9y\right]+\left[1\right]{e}^{9y}\\ & =x{e}^{9y}\left[9\frac{dy}{dx}\right]+{e}^{9y}\\ & =9x{e}^{9y}{y}^{\prime }+{e}^{9y}\end{array}$
Now do the same thing for the second term, then solve for y′

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