Question

F(x) = 2x ^ 2 - 3x ^ 2 + 2x - 1

Answer 1

1a)

`\begin{gathered} f(x)=2x^3-3x^2+2x-1 \\ \Rightarrow f^(\prime)(x)=2\cdot3x^(3-1)-3\cdot2x^(2-1)+2x^(1-1)=6x^2-6x+2^{} \\ \Rightarrow f^(\prime)(x)=6x^2-6x+2 \end{gathered}`

1b)

`G(t)=2\sqrt[]{t}+\frac{3}{\sqrt[]{t}}=2t^{(1)/(2)}+3t^{-(1)/(2)}`
`\begin{gathered} \Rightarrow G^(\prime)(t)=(1)/(2)\cdot2t^{(1)/(2)-1}+3(-(1)/(2))t^{-(1)/(2)-1}=t^{-(1)/(2)}-(3)/(2)t^{-(3)/(2)} \\ \Rightarrow G^(\prime)(t)=\frac{1}{\sqrt[]{t}}-\frac{3}{2\sqrt[2]{t^3}} \end{gathered}`

1c)

`g(t)=(4)/(t^4)-(3)/(t^3)+(2)/(t)=4t^(-4)-3t^(-3)+2t^(-1)`
`\begin{gathered} \Rightarrow g^(\prime)(t)=4(-4)t^(-4-1)-3(-3)t^(-3-1)+2(-1)t^(-1-1)=-16t^(-5)+9t^(-4)-2t^(-2) \\ \Rightarrow g^(\prime)(t)=-(16)/(t^5)+(9)/(t^4)-(2)/(t^2)^{} \end{gathered}`

1d)

`f(x)=(3)/(x^3)+\frac{4}{\sqrt[]{x}}+1=3x^(-3)+4x^{-(1)/(2)}+1`
`\begin{gathered} \Rightarrow f^(\prime)(x)=3(-3)x^(-3-1)+4(-(1)/(2))x^{-(1)/(2)-1}=-9x^(-4)-2x^{-(3)/(2)}=-(9)/(x^4)-\frac{2}{\sqrt[2]{x^3}^{}} \\ \Rightarrow f^(\prime)(x)=-(9)/(x^4)-\frac{2}{\sqrt[2]{x^3}^{}} \end{gathered}`

1e)

`f(x)=(x^3+2x^2+x-1)/(x)=x^2+2x+1-(1)/(x)=x^2+2x+1-x^(-1)`
`\begin{gathered} \Rightarrow f^(\prime)(x)=2x+2-(-1)x^(-1-1)=2x+2+x^(-2)=2x+2+(1)/(x^2) \\ \Rightarrow f^(\prime)(x)=2x+2+(1)/(x^2) \end{gathered}`