Question
Find \( \frac{d y}{d x} \) by implicit differentiation. \[ x^{8}+y^{5}=4 \] \( \frac{d y}{d x}=\square \)
Ask by Rowe Tucker. in the United States
Mar 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\frac{dy}{dx} = -\frac{8x^7}{5y^4}
\]
Solution
Differentiate both sides of
\[
x^8 + y^5 = 4
\]
with respect to \( x \).
1. The derivative of \( x^8 \) is
\[
8x^7.
\]
2. The derivative of \( y^5 \) with respect to \( x \) requires the chain rule:
\[
\frac{d}{dx}(y^5) = 5y^4 \frac{dy}{dx}.
\]
3. The derivative of the constant \( 4 \) is \( 0 \).
Thus, we have:
\[
8x^7 + 5y^4 \frac{dy}{dx} = 0.
\]
Solve for \( \frac{dy}{dx} \):
Subtract \( 8x^7 \) from both sides:
\[
5y^4 \frac{dy}{dx} = -8x^7.
\]
Divide both sides by \( 5y^4 \):
\[
\frac{dy}{dx} = -\frac{8x^7}{5y^4}.
\]
The final answer is:
\[
\frac{dy}{dx} = -\frac{8x^7}{5y^4}.
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution The Deep Dive
To find \( \frac{d y}{d x} \) using implicit differentiation, we start with the equation: \[ x^{8} + y^{5} = 4. \] Differentiate both sides with respect to \( x \): \[ \frac{d}{dx}(x^8) + \frac{d}{dx}(y^5) = \frac{d}{dx}(4). \] This gives us: \[ 8x^{7} + 5y^{4} \frac{dy}{dx} = 0. \] Now, solve for \( \frac{dy}{dx} \): \[ 5y^{4} \frac{dy}{dx} = -8x^{7}, \] \[ \frac{dy}{dx} = -\frac{8x^{7}}{5y^{4}}. \] Thus, the derivative \( \frac{d y}{d x} \) is: \[ \frac{d y}{d x} = -\frac{8x^{7}}{5y^{4}}. \]
