Question

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

Answer 1

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).