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Chapter 3: Problem 50

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. $$x y^{5 / 2}+x^{3 / 2} y=12 ;(4,1)$$

Short Answer

Expert verified
Question: Find the slope of the curve at the given point (4,1) for the equation $x y^{5 / 2}+x^{3 / 2} y = 12$. Answer: The slope of the curve at the given point (4,1) is $-\frac{3}{7}$.

Step by step solution

01

Apply Implicit Differentiation

Start by differentiating both sides of the given equation with respect to x: $$\frac{d}{dx}(x y^{5 / 2}+x^{3 / 2} y) = \frac{d}{dx}(12)$$ Now apply the chain rule for implicit differentiation to each term.
02

Differentiate each term

Differentiating the first term with respect to x: $$\frac{d}{dx}(x y^{5 / 2}) = y^{5 / 2} + x \frac{d}{dx}(y^{5 / 2}) = y^{5 / 2} + \frac{5}{2} x^{1 / 2} y^{3 / 2} \frac{dy}{dx}$$ Differentiating the second term with respect to x: $$\frac{d}{dx}(x^{3 / 2} y) = x^{1 / 2} y + \frac{3}{2} x^{-1 / 2} y \frac{dy}{dx}$$ Combined, we have: $$y^{5 / 2} + \frac{5}{2} x^{1 / 2} y^{3 / 2} \frac{dy}{dx} + x^{1 / 2} y + \frac{3}{2} x^{-1 / 2} y \frac{dy}{dx} = 0$$
03

Solve for the derivative dy/dx

We need to solve the equation above to find the derivative. First, isolate the terms containing dy/dx: $$\frac{dy}{dx}(\frac{5}{2} x^{1 / 2} y^{3 / 2} + \frac{3}{2} x^{-1 / 2} y) = -y^{5 / 2} - x^{1 / 2} y$$ Now, divide both sides by the terms within the parentheses to get dy/dx: $$\frac{dy}{dx} = \frac{-y^{5/2} - x^{1/2}y}{\frac{5}{2} x^{1/2} y^{3/2} + \frac{3}{2} x^{-1/2} y}$$
04

Evaluate the derivative at the given point

Plug in the given point (4,1) into the derivative equation: $$\frac{dy}{dx}(4,1) = \frac{-(1)^{5 / 2} - (4)^{1 / 2}(1)}{\frac{5}{2} (4)^{1 / 2} (1)^{3 / 2} + \frac{3}{2} (4)^{-1 / 2} (1)}$$ Simplify the expression: $$\frac{dy}{dx}(4,1) = \frac{-(1) - 2}{\frac{5}{2}(2) + \frac{3}{2}} = \frac{-3}{7}$$ Therefore, the slope of the curve at the given point (4,1) is: $$\frac{dy}{dx}(4,1) = -\frac{3}{7}$$

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Most popular questions from this chapter

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises \(63-68 .)\) b. Graph the tangent and normal lines on the given graph. $$\begin{aligned}&3 x^{3}+7 y^{3}=10 y\\\&\left(x_{0}, y_{0}\right)=(1,1)\end{aligned}$$

Find the slope of the curve \(5 \sqrt{x}-10 \sqrt{y}=\sin x\) at the point \((4 \pi, \pi)\).

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{f(x) g(x)}{x}\right]\right|_{x=4}$$

Means and tangents Suppose \(f\) is differentiable on an interval containing \(a\) and \(b,\) and let \(P(a, f(a))\) and \(Q(b, f(b))\) be distinct points on the graph of \(f\). Let \(c\) be the \(x\) -coordinate of the point at which the lines tangent to the curve at \(P\) and \(Q\) intersect, assuming that the tangent lines are not parallel (see figure). a. If \(f(x)=x^{2},\) show that \(c=(a+b) / 2,\) the arithmetic mean of \(a\) and \(b\), for real numbers \(a\) and \(b\) b. If \(f(x)=\sqrt{x},\) show that \(c=\sqrt{a b},\) the geometric mean of \(a\) and \(b,\) for \(a>0\) and \(b>0\) c. If \(f(x)=1 / x,\) show that \(c=2 a b /(a+b),\) the harmonic mean of \(a\) and \(b,\) for \(a>0\) and \(b>0\) d. Find an expression for \(c\) in terms of \(a\) and \(b\) for any (differentiable) function \(f\) whenever \(c\) exists.

Witch of Agnesi Let \(y\left(x^{2}+4\right)=8\) (see figure). a. Use implicit differentiation to find \(\frac{d y}{d x}\) b. Find equations of all lines tangent to the curve \(y\left(x^{2}+4\right)=8\) when \(y=1\) c. Solve the equation \(y\left(x^{2}+4\right)=8\) for \(y\) to find an explicit expression for \(y\) and then calculate \(\frac{d y}{d x}\) d. Verify that the results of parts (a) and (c) are consistent.
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