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Chapter 3: Problem 45

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. $$\sqrt[3]{x}+\sqrt[3]{y^{4}}=2 ;(1,1)$$

Short Answer

Expert verified
Answer: The slope of the curve at the given point \((1,1)\) is \(-\frac{1}{4}\).

Step by step solution

01

Differentiate both sides with respect to x

Differentiate the given equation with respect to \(x\): $$\frac{d}{dx}\left(\sqrt[3]{x}+\sqrt[3]{y^4}\right) = \frac{d}{dx}(2)$$
02

Solve for dy/dx

Apply the differentiation rules. For the term \(\sqrt[3]{y^4}\), apply the chain rule with respect to \(y\) and then multiply by \(\frac{dy}{dx}\): $$\frac{1}{3}x^{-\frac{2}{3}} + \frac{4}{3}y^{\frac{1}{3}}\frac{dy}{dx} = 0$$ To find the slope of the curve at the point \((1,1)\), we substitute the given coordinates into the equation: $$\frac{1}{3}(1)^{-\frac{2}{3}} + \frac{4}{3}(1)^{\frac{1}{3}}\frac{dy}{dx} = 0$$ Now, we can solve for \(\frac{dy}{dx}\): $$\frac{dy}{dx} = -\frac{1}{4}(1)^{\frac{2}{3}} = -\frac{1}{4}$$
03

Evaluate the slope at the given point

Now that we have found the slope, we can evaluate the expression at the given point \((1,1)\): $$\frac{dy}{dx}(1,1) = -\frac{1}{4}$$ Therefore, the slope of the curve at the given point \((1,1)\) is \(\boxed{-\frac{1}{4}}\).

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