# Practice Problems for Module 6

Sections 3.3–3.6

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1. Differentiate the functions.
1. $$F(x)=\left(1+x+x^2\right)^{200}$$
2. $$g(\theta)=\cos^2\theta$$

1. $$F'(x)=200\left(1+x+x^2\right)^{199}(1+2x)$$
2. $$g'(\theta)=-2\cos\theta \sin\theta$$

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3. $$y=e^{\tan \theta}$$
4. $$F(t)=e^{t\sin 2t}$$

1. $$y'=e^{\tan\theta}\sec^2\theta$$
2. $$F'(t)=e^{t\sin(2t)}\left(\sin(2t)+2t\cos(2t)\right)$$

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5. $$f(t)=\tan (\sec (\cos t))$$

$$f'(t)=\sec^2(\sec(\cos t))\sec(\cos t)\tan(\cos t)(-\sin t)$$

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2. Find $$y'$$ and $$y''$$ for $$y=e^{e^x}$$

$$y'=e^{e^x}e^x$$
$$y''=e^{e^x+x}\left(e^x+1\right)$$

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3. Let $$r(x)=f(g(h(x)))$$, where $$h(1)=2$$, $$g(2)=3$$, $$h'(1)=4$$, $$g'(2)=5$$, and $$f'(3)=6$$. Find $$r'(1)$$.

$$r'(1)=120$$

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4. If $$g(x)=f(3f(4f(x)))$$, where $$f(0)=0$$ and $$f'(0)=2$$, find $$g'(0)$$.

$$g'(0)=96$$

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5. Find the $$2020$$th derivative of $$y=\cos 2x$$.

$$y^{(2020)}=2^{2020}\cos(2x)$$

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6. Find the $$2020$$th derivative of $$f(x)=xe^{-x}$$.

$$f^{(2020)}(x)=-(2020-x)e^{-x}$$

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7. If $$f$$ and $$g$$ are the functions whose graphs are shown, let $$u(x)=f(g(x))$$, $$v(x)=g(f(x))$$, and $$w(x)=g(g(x))$$. Find $$u'(1)$$, $$v'(1)$$, and $$w'(1)$$.

$$u'(1)=\dfrac{3}{4}$$
$$v'(1)=DNE$$
$$w'(1)=-2$$

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8. Find $$\dfrac{dy}{dx}$$.
1. $$x^3-xy^2+y^3=1$$.

$$\dfrac{dy}{dx}=\dfrac{-3x^2+y^2}{-2xy+3y^2}$$

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2. $$\cos(xy)=1+\sin y$$.

$$\dfrac{dy}{dx}=\dfrac{-y\sin(xy)}{\cos y +x\sin(xy)}$$

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3. $$e^y\sin x = x + xy$$.

$$\dfrac{dy}{dx}=\dfrac{1+y-e^y\cos x}{e^y \sin x -x}$$

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4. $$x\sin y + y \sin x = 1$$.

$$\dfrac{dy}{dx}=\dfrac{-\sin y - y \cos x}{x\cos y + \sin x}$$

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9. ​If $$g(x)+x \sin ( g(x))=x^2$$, find $$g'(x)$$.

$$g'(x)=\dfrac{2x-\sin (g(x))}{1+x\cos(g(x))}$$

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10. Find an equation of the tangent line to the curve at the given point.
1. $$x^2+2xy+4y^2=12, \quad (2,1)$$

$$y=-\dfrac{1}{2}x+2$$

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2. $$y\sin 2x = x\cos 2y, \quad \left(\dfrac{\pi}{2},\dfrac{\pi}{4}\right)$$

$$y=\dfrac{1}{2}x$$

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11. ​Find the derivative of the function:
1. $$y=\left(\tan^{-1} x\right)^2$$
2. $$y=\tan^{-1}\left(x^2\right)$$​

1. $$\dfrac{dy}{dx}=2\left(\tan^{-1} x\right) \dfrac{1}{1+x^2}$$
2. $$\dfrac{dy}{dx}=\dfrac{2x}{1+x^4}$$

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3. $$R(t)=\arcsin\left(\dfrac{1}{t}\right)$$
4. $$f(x)=\arctan \left(x^2-x\right)$$

1. $$\dfrac{dR}{dt}=\dfrac{-1}{t^2\sqrt{1-\dfrac{1}{t^2}}}$$
2. $$\dfrac{df}{dx}=\dfrac{2x-1}{1+\left(x^2-x\right)^2}$$

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5. $$f(x)=\ln (\sin^2 x)$$
6. $$g(x)=\ln \left(xe^{-2x}\right)$$

1. $$f'(x)=\dfrac{2\cos x}{\sin x}$$
2. $$g'(x)=\dfrac{1-2x}{x}$$

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7. $$f(x)=\log(1+\cos x)$$
8. $$F(s)=\ln \ln s$$
9. $$y=\log_2\left(x\log_5 x\right)$$

1. $$f'(x)=\dfrac{-\sin x}{(1+\cos x)\ln 10}$$
2. $$F'(s)=\dfrac{1}{s\ln s}$$
3. $$y'=\dfrac{\log_5 x + \dfrac{1}{\ln 5}}{\left(x\log_5 x\right)\ln 2}$$

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12. If $$f(x)=\cos \left(\ln x^2\right)$$

$$f'(1)=0$$

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13. Find the equation of the tangent line to the curve $$y=x^2\ln x$$ at the point $$(1,0)$$.

$$y=x-1$$

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14. Use the logarithmic differentiation to find the derivative of the function.
1. $$y=\dfrac{e^{-x}\cos^2x}{x^2+x+1}$$.

$$y'=\dfrac{e^{-x}\cos^2x}{x^2+x+1}\left(-1-\dfrac{2\sin x}{\cos x} - \dfrac{2x+1}{x^2+x+1}\right)$$

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2. $$y=x^x$$

$$y'=x^2(\ln x + 1)$$

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3. $$y=(\ln x)^{\cos x}$$

$$y'=(\ln x)^{\cos x} \left( -\sin x \cdot \ln (\ln x) + \dfrac{\cos x}{x\ln x}\right)$$

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