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a:\(\Delta y=\left\lbrack2\left(x+\Delta x\right)-5\right\rbrack-\left(2x-5\right)\)
\(=2\left(x+\Delta x\right)-2x\)
\(=2\cdot\Delta x\)
=>\(\frac{\Delta y}{\Delta x}=2\)
b: \(\Delta y=\left\lbrack\left(x+\Delta x\right)^2-1\right\rbrack-\left(x^2-1\right)\)
\(=\left(x+\Delta x\right)^2-x^2=\left(x+\Delta x-x\right)\left(x+\Delta x+x\right)=\Delta x\left(2x+\Delta x\right)\)
=>\(\frac{\Delta y}{\Delta x}=2x+\Delta x\)
c: \(y=2x^3\)
=>\(\Delta y=2\left(x+\Delta x\right)^3-2x^3\)
\(=2\left(x^3+3\cdot x^2\cdot\Delta x+3\cdot x\cdot\left(\Delta x\right)^2+\left(\Delta x\right)^3\right)-2x^3\)
\(=2\left(3x^2\cdot\Delta x+3\cdot x\cdot\left(\Delta x\right)^2+\left(\Delta x\right)^3\right)\)
=>\(\frac{\Delta y}{\Delta x}=\frac{2\cdot\Delta x\left(3x^2+3x+\left(\Delta x\right)^2\right)}{\Delta x}=2\left\lbrack3x^2+3x+\left(\Delta x\right)^2\right\rbrack\)
d: \(y=\frac{1}{x}\)
=>\(\Delta y=\frac{1}{x+\Delta x}-\frac{1}{x}=\frac{x-x-\Delta x}{x\left(x+\Delta x\right)}=\frac{-\Delta x}{x\left(x+\Delta x\right)}\)
=>\(\frac{\Delta y}{\Delta x}=\frac{-1}{x\left(x+\Delta x\right)}\)
