Question

In each of the following evaluate y' when x =2 :

1)$8x-x^3$

2)$a{x}^2+bx+c$

3)$x+a^3$

4)$\sqrt{2}x$

Answer 1

Differentiating wrt x [using $\frac{d{x}^n}{dn}=n{x}^{n-1}]$

$\frac{d}{dx}(8x-x^3{)}_{x=2}=(8-3x^2{)}_{x=2}=8-12=-4$

$\frac{d}{dx}(a{x}^2+bx+c{)}_{x=2}=(2ax+b{)}_{x=2}=4a+b$

$\frac{d}{dx}(x+a^3{)}_{x=2}=1=1$

$\frac{d}{dx}1/\sqrt{2x}=\frac{d}{dx}\left(\frac{1}{\sqrt{2}}{x}^{-1/2}\right)=(\frac{1}{\sqrt{2}}\times \frac{-1}{2}\times x^{-3/2}{)}_{x=2}$

$=\frac{-1}{4_{2\sqrt{2}}}{2}^{-3/2}=\frac{-1}{4\times 2}=\frac{-1}{8}$