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From first principles, find the indicated derivatives​

From First Principles Find The Indicated Derivatives class=

Sagot :

LammettHash

By definition of the derivative,

[tex]\displaystyle\frac{dr}{ds} = \lim_{h\to0} \frac{\left(\frac{(s + h)^3}2 + 1\right) - \left(\frac{s^3}2 + 1\right)}{h}[/tex]

[tex]\displaystyle\frac{dr}{ds} = \lim_{h\to0} \frac{\left(\frac{s^3+3s^2h+3sh^2+h^3}2 + 1\right) - \left(\frac{s^3}2 + 1\right)}{h}[/tex]

[tex]\displaystyle\frac{dr}{ds} = \lim_{h\to0} \frac{\frac{3s^2h+3sh^2+h^3}2}{h}[/tex]

[tex]\displaystyle\frac{dr}{ds} = \lim_{h\to0} \frac12 \frac{3s^2h+3sh^2+h^3}{h}[/tex]

[tex]\displaystyle\frac{dr}{ds} = \lim_{h\to0} \frac12 (3s^2+3sh+h^2)[/tex]

[tex]\displaystyle\frac{dr}{ds} = \frac{3s^2}2[/tex]

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