answers:
a. dy/dx=−x⁴/y⁴
b. dy/dx=−1 at (3,−3)
steps
Sure, here are the steps on how to find the slope of the curve x⁵+y⁵=0 at the point (3,−3):
**a. Using implicit differentiation to find dx/dy**
We can use implicit differentiation to find dx/dy as follows:
1. Start by differentiating both sides of the equation x⁵+y⁵=0 with respect to x.
2. When we differentiate y, we need to use the chain rule.
3. We will get an expression for dx/dy on the left-hand side of the equation.
Here is the differentiation process:
```
x⁵+y⁵=0
d(x⁵+y⁵)=d(0)
5x⁴+5y⁴dy/dx=0
5x⁴+5y⁴dy/dx=0
dy/dx=−5x⁴/5y⁴
dy/dx=−x⁴/y⁴
```
Therefore, the expression for dx/dy is **dy/dx=−x⁴/y⁴**.
**b. Finding the slope of the curve at the given point x⁵+y⁵=0; (3,−3)**
To find the slope of the curve at the given point, we need to substitute the values of x and y into the expression for dx/dy.
The given point is (3,−3), so we have:
```
dy/dx=−x⁴/y⁴
dy/dx=−(3)⁴/(−3)⁴
dy/dx=−81/81
dy/dx=−1
```
Therefore, the slope of the curve at the given point is **-1**.
To summarize, the steps on how to find the slope of the curve x⁵+y⁵=0 at the point (3,−3) are as follows:
1. Use implicit differentiation to find dx/dy.
2. Substitute the values of x and y into the expression for dx/dy.
3. Simplify the expression to obtain the slope of the curve.
The slope of the curve at the given point is **-1**.
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