QuestionAugust 27, 2025

Use implicit differentiation to find the derivative at (1,2) 10x^4+y^4=26 slope=(-[?])/([ ])

Use implicit differentiation to find the derivative at (1,2) 10x^4+y^4=26 slope=(-[?])/([ ])
Use implicit differentiation to find the derivative at (1,2) 10x^4+y^4=26 slope=(-[?])/([ ])

Solution
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Answer

slope = \frac{-5}{4} Explanation 1. Differentiate both sides with respect to x Differentiate 10x^4 + y^4 = 26 implicitly. The derivative of 10x^4 is 40x^3. For y^4, use the chain rule: 4y^3 \frac{dy}{dx}. 2. Set up the equation 40x^3 + 4y^3 \frac{dy}{dx} = 0. 3. Solve for \frac{dy}{dx} Rearrange to find \frac{dy}{dx}: \frac{dy}{dx} = -\frac{40x^3}{4y^3}. 4. Simplify the expression Simplify to get \frac{dy}{dx} = -\frac{10x^3}{y^3}. 5. Substitute (1,2) into the derivative Substitute x = 1 and y = 2: \frac{dy}{dx} = -\frac{10(1)^3}{(2)^3} = -\frac{10}{8} = -\frac{5}{4}.

Explanation

1. Differentiate both sides with respect to $x$<br /> Differentiate $10x^4 + y^4 = 26$ implicitly. The derivative of $10x^4$ is $40x^3$. For $y^4$, use the chain rule: $4y^3 \frac{dy}{dx}$.<br />2. Set up the equation<br /> $40x^3 + 4y^3 \frac{dy}{dx} = 0$.<br />3. Solve for $\frac{dy}{dx}$<br /> Rearrange to find $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{40x^3}{4y^3}$.<br />4. Simplify the expression<br /> Simplify to get $\frac{dy}{dx} = -\frac{10x^3}{y^3}$.<br />5. Substitute $(1,2)$ into the derivative<br /> Substitute $x = 1$ and $y = 2$: $\frac{dy}{dx} = -\frac{10(1)^3}{(2)^3} = -\frac{10}{8} = -\frac{5}{4}$.
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