Question
Given the equation \( x-y^{8}=3 \), find \( \frac{d x}{d y} \) by implicit differentiation.
Ask by Hall Morgan. in the United States
Mar 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \frac{dx}{dy} = 8y^7 \)
Solution
1. Differentiate both sides of the equation with respect to \( y \):
\[
\frac{d}{dy}(x) - \frac{d}{dy}(y^8) = \frac{d}{dy}(3)
\]
2. Compute each derivative:
- For \( x \), since \( x \) is a function of \( y \):
\[
\frac{d}{dy}(x) = \frac{dx}{dy}
\]
- For \( y^8 \):
\[
\frac{d}{dy}(y^8) = 8y^7
\]
- For the constant \( 3 \):
\[
\frac{d}{dy}(3) = 0
\]
3. Substitute the derivatives back into the equation:
\[
\frac{dx}{dy} - 8y^7 = 0
\]
4. Solve for \( \frac{dx}{dy} \):
\[
\frac{dx}{dy} = 8y^7
\]
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Beyond the Answer
To find \( \frac{dx}{dy} \) using implicit differentiation, we start with the given equation: \[ x - y^8 = 3 \] Differentiating both sides with respect to \( y \): \[ \frac{d}{dy}(x) - \frac{d}{dy}(y^8) = \frac{d}{dy}(3) \] This gives us: \[ \frac{dx}{dy} - 8y^7 = 0 \] Now, solving for \( \frac{dx}{dy} \): \[ \frac{dx}{dy} = 8y^7 \] Thus, the derivative \( \frac{dx}{dy} \) is: \[ \frac{dx}{dy} = 8y^7 \]