# What is the implicit derivative of #1= xe^(4y#?

Product Rule:

Find each derivative separately:

Plug them back in:

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To find the implicit derivative of the equation (1 = xe^{4y}), differentiate both sides of the equation with respect to (x) using the chain rule.

The implicit derivative is:

[\frac{dy}{dx} = -\frac{f'(x)}{f'(y)}]

where (f(x, y) = 1 - xe^{4y}).

First, find (\frac{df}{dx}) and (\frac{df}{dy}):

[\frac{df}{dx} = -e^{4y}] [\frac{df}{dy} = -4xe^{4y}]

Now, apply the formula:

[\frac{dy}{dx} = -\frac{f'(x)}{f'(y)} = -\frac{-e^{4y}}{-4xe^{4y}} = \frac{e^{4y}}{4xe^{4y}} = \frac{1}{4x}]

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