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Find (dy)(d x) where y⁸e⁻³ˣ+10x²-y⁹=-4.

User Kumar
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Final Answer:

The derivative
\((dy)/(dx)\) for the given expression
\(y^8e^(-3x) + 10x^2 - y^9 = -4\) is determined as follows:


\[ (dy)/(dx) = (-2y^8e^(-3x) + 20x)/(8y^7e^(-3x) + 20x) \]

Step-by-step explanation:

The given expression is
\( y^8e^(-3x) + 10x^2 - y^9 = -4 \). To find
\((dy)/(dx)\), we'll

differentiate both sides of the equation with respect to (x).

First, let's focus on the left side. Using the chain rule for the term
\(y^8e^(-3x)\),

we get
\(8y^7e^(-3x) (dy)/(dx) - 3y^8e^(-3x)\). For the term
\(10x^2\), the derivative is
\(20x\). For

the term
\(-y^9\), the derivative is
\(-9y^8 (dy)/(dx)\). So, the left side

becomes
\(8y^7e^(-3x) (dy)/(dx) - 3y^8e^(-3x) + 20x - 9y^8 (dy)/(dx)\).

Setting this equal to the derivative of the constant term (-4), which is (0), we have the following equation:


\[8y^7e^(-3x) (dy)/(dx) - 3y^8e^(-3x) + 20x - 9y^8 (dy)/(dx) = 0\]

Now, rearrange and solve for
\((dy)/(dx)\):


\[ (dy)/(dx) = (-2y^8e^(-3x) + 20x)/(8y^7e^(-3x) + 20x) \]

This is the final expression for
\((dy)/(dx)\) in terms of (x) and (y). The numerator and denominator terms are obtained through the differentiation and rearrangement process, resulting in the derivative of (y) with respect to (x).

User Roman Kazmin
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