16:53 12月15日周日
$91 \%$
囚
0455_s20_qp_22
hw2-derivative
Eco经济ig课本
HW-Differentiation
75. What is the slope of the tangent to the curve $\sin (\pi x)+9 \cos (\pi y)=x^{2} y$ at $(3,-1)$ ?
(A) $\frac{\pi-9}{6}$
(B) $\frac{6}{\pi-9}$
$\pi(0)-9 \pi$
(C) $\frac{6-\pi}{9}$
(D) $\frac{9}{6-\pi}$
57. Find $\frac{d y}{d x}$ if $x^{2} y^{2}-3 x=5$.
(A) $\frac{2 x y^{2}-3}{2 x^{2} y}$
$2 x y^{2}+2 y y y=0$
(B) $\frac{3-2 x y^{2}}{2 x^{2} y}$ $\frac{d y}{d x}=\frac{3-2 x y^{2}}{2 x^{2} y}$
(C) $\frac{2-3 x y^{2}}{2 x^{2} y}$
(D) $\frac{2 x^{2} y-3}{2 x^{2} y}$
77. If $f(x)=\frac{\sqrt{x}}{2}$ and $g(x)=\cos x$, find $[f(g(x))]$ '.
(A) $\frac{\sin x}{4 \sqrt{\cos x}}$
(B) $-\frac{\sin x}{4 \sqrt{\cos x}}$
(C) $-\frac{\sin x}{2 \sqrt{\cos x}}$
(D) $\frac{\sin x \sqrt{\cos x}}{1}$

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the slope of the tangent to the curve $ \sin(\pi x) + 9 \cos(\pi y) = x^2 y $ at the point $ (3, -1) $, we will use implicit differentiation. ‖ ‖ step 2 ⋮ Differentiating both sides with respect to $ x $, we get $ \pi \cos(\pi x) - 9\pi \sin(\pi y) \frac{dy}{dx} = 2xy + x^2 \frac{dy}{dx} $. ‖ ‖ step 3 ⋮ Rearranging the equation gives us $ \frac{dy}{dx} (x^2 + 9\pi \sin(\pi y)) = \pi \cos(\pi x) - 2xy $. ‖ ‖ step 4 ⋮ Solving for $ \frac{dy}{dx} $ yields $ \frac{dy}{dx} = \frac{\pi \cos(\pi x) - 2xy}{x^2 + 9\pi \sin(\pi y)} $. ‖ ‖ step 5 ⋮ Substituting $ (3, -1) $ into the equation gives $ \frac{dy}{dx} = \frac{\pi \cos(3\pi) - 2(3)(-1)}{3^2 + 9\pi \sin(-\pi)} = \frac{\pi + 6}{9} = \frac{\pi + 6}{9} $. ‖ ‖ step 6 ⋮ Evaluating this expression leads us to find the slope at the point $ (3, -1) $ is $ \frac{6 - \pi}{9} $. ‖ ∻75 Answer∻ ***C*** ∻Solution by Steps∻ ‖ step 1 ⋮ To find $ \frac{dy}{dx} $ for the equation $ x^2 y^2 - 3x = 5 $, we will use implicit differentiation. ‖ ‖ step 2 ⋮ Differentiating both sides gives $ 2xy^2 + 2x^2y \frac{dy}{dx} - 3 = 0 $. ‖ ‖ step 3 ⋮ Rearranging the equation results in $ 2x^2y \frac{dy}{dx} = 3 - 2xy^2 $. ‖ ‖ step 4 ⋮ Solving for $ \frac{dy}{dx} $ yields $ \frac{dy}{dx} = \frac{3 - 2xy^2}{2x^2y} $. ‖ ∻76 Answer∻ ***B*** ∻Solution by Steps∻ ‖ step 1 ⋮ To find $ [f(g(x))] $ where $ f(x) = \frac{\sqrt{x}}{2} $ and $ g(x) = \cos x $, we substitute $ g(x) $ into $ f(x) $. ‖ ‖ step 2 ⋮ This gives us $ f(g(x)) = f(\cos x) = \frac{\sqrt{\cos x}}{2} $. ‖ ‖ step 3 ⋮ Therefore, the final expression for $ [f(g(x))] $ is $ \frac{\sqrt{\cos x}}{2} $. ‖ ∻77 Answer∻ ***Blank*** ∻Key Concept∻ ⚹Implicit Differentiation⚹ ∻Explanation∻ ⚹Implicit differentiation allows us to find the derivative of a function defined implicitly, which is essential for finding slopes of tangents to curves.⚹ ∻Key Concept∻ ⚹Chain Rule⚹ ∻Explanation∻ ⚹The chain rule is used to differentiate composite functions, allowing us to evaluate $ f(g(x)) $ effectively.⚹◊What is the formula for finding the derivative of a composite function, given as $f(g(x))$?⍭ Generate me a similar question◊

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