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Find the Derivative: $$f(x)=8.6 \quad f(x)=x^{2} \quad F(x)=2 x^{3}-x^{2}+6$$

$$f(x)=8.6 \longrightarrow f^{\prime}(x)=0$$

$$f(x)=x^{2} \longrightarrow f^{\prime}(x)=2 x^{2-1}=2 x$$

$$f(x)=2 x^{3}-x^{2}+6 \longrightarrow f^{\prime}(x)=(2)(3) x^{3-1}-2 x^{2-1}+0$$

$$f^{\prime}(x)=6 x^{2}-2 x$$

Find the derivative for the following functions:
$$\begin{array}{ll}{\text { (a) } f(x)=186.5} & {\text { (b) } f(x)=x^{3}} \\ {(c) f(x)=-4 x^{10}} & {\text { (d) } A(s)=\frac{-12}{s^{5}}} \\ {\text { (e) } f(x)=1.4 t^{5}-2.5 t^{2}+6.7} & {\text { (f) } y=\frac{x^{2}+4 x+3}{\sqrt{x}}}\end{array}$$

(a) $$f(x)=186.5 \Rightarrow f^{\prime}(x)=0$$

(b) $$f(x)=x^{3} \Rightarrow 3 x^{3-1}=3 x^{2}$$

(c) $$f(x)=-4 x^{(10)} \Rightarrow-4(10) x^{10-1}=-40 x^{9}$$

(d) $$A(s)=\frac{-12}{s^{5}} =-12 {s}^{-5} \Rightarrow$$

$$-12(-5) s^{-5-1}=60s^{-6}=\frac{60}{s^{6}}$$

(e) $$f(x)=1.4 t^{5}-2.5 t^{2}+6.7=1.4(5) t^{4}-2.5(2) t+0$$

$$f(x)=7 t^{4}-5 t$$

(f) $$y=\frac{x^{2}+4 x+3}{\sqrt{x}}=\frac{x^{2}+4 x+3}{x^{1 / 2}}$$

$$y=\left(x^{2}+4 x^{1}+3\right) x^{-1 / 2}=x^{1.5}+4 x^{0.5}+3 x^{-0.5}$$

$$y^{\prime}=1.5 x^{0.5}+4(0.5) x^{-0.5}+3(-0.5) x^{-1.5}$$

$$y^{\prime}=1.5 x^{0.5}+2 x^{-0.5}+-1.5 x^{-1.5}$$

$$=1.5 \sqrt{x}+\frac{2}{\sqrt{x}}-\frac{1.5}{\sqrt{x^{3}}}$$

Find the Derivative: $$F(x)=\left(x^{3}+2 x\right) \mathrm{e}^{x} \quad g(x)=\frac{3 x-1}{2 x+1} \quad g(t)=t^{3} \cos t$$

$$F(x)=\left(x^{3}+2 x\right) e^{x} \longrightarrow\left(3 x^{2}+2\right) e^{x}+e^{x}\left(x^{3}+2 x\right)$$

$$(f \cdot g)^{\prime}=f^{\prime} \cdot g+g^{\prime} f$$

$$g(x)=\frac{3 x-1}{2 x+1} \rightarrow g^{\prime}(x)=\frac{(3)(2 x+1)-(2)(3 x-1)}{(2 x+1)^{2}}$$

$$g^{\prime}(x)=\frac{6 x+3-6 x+2}{(2 x+1)^{2}}=\frac{5}{(2 x+1)^{2}}$$

$$g(t)=t^{3} \cdot \cos t=3 t^{2} \cos t+t^{3}(-\sin t)=3 t^{2} \cos t-t^{3} \sin t$$

Find the Derivative: $$y=\frac{x}{2-\tan x}$$

$$y=\frac{x}{2-\tan x}=\frac{(1)(2-\tan x)-(-\sec^{2} x)(x)}{(2-\tan x)^{2}}$$

$$y^{\prime}=\frac{2-\tan x+x \sec ^{2} x}{(2-\tan x)^{2}}$$

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