 # Practice Problems for Module 5

Sections 3.1 and 3.2

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1. Differentiate the functions.​
1. ​$$f(x)=\dfrac{7}{4}x^4-3x^2+12$$
2. $$H(u)=(3u-1)(u+5)$$

1. $$f'(x)=7x^3-6x$$
2. $$H'(u)=3u^2+14u-5$$

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3. $$F(r)=\dfrac{8}{r^3}$$​
4. $$y=\dfrac{\sqrt{x}+x}{x^2}$$​

1. $$F'(r)=-24r^{-4}$$
2. $$y'=-\dfrac{5}{3}x^{-8/3}-x^{-2}$$

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5. $$k(x)=e^x+x^e$$​​

$$k'(x)=e^x+ex^{e-1}$$

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6. $$f(x)=(3x^2-x)e^x$$​​

$$f'(x)=(6x-1)e^x+(3x^2-x)e^x$$

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7. $$y=\dfrac{e^x}{4-e^x}$$​

$$y'=\dfrac{e^x(4-e^x)-e^x(-e^x)}{\left(4-e^x\right)^2}$$

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8. $$G(x)=\dfrac{x^2-3}{5x+1}$$​​

$$G'(x)=\dfrac{(2x)(5x+1)-(x^2-3)(5)}{(5x+1)^2}$$

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9. $$F(y)=\left( \dfrac{1}{y^2}-\dfrac{3}{y^4}\right)\left(y+2y^3\right)$$​​

$$F'(y)=\left(-2y^{-3}+12y^{-5}\right)\left(y+2y^3\right)+\left(y^{-2}-3y^{-4}\right)\left(1+6y^2\right)$$

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10. $$V(t)=\dfrac{7+t}{te^t}$$​​

$$V'(t)=\dfrac{(1)\left(te^t\right)-(7+t)\left(e^t+te^t\right)}{\left(te^t\right)^2}$$

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2. Find $$f'(x)$$ and $$f''(x)$$ for $$f(x)=\left(x^3+1\right)e^x$$.

$$f'(x)=\left(x^3+3x^2+1\right)e^x$$
$$f''(x)=\left(3x^2+6x\right)e^x +\left(x^3+3x^2+1\right)e^x$$

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