Hi!

Apparently I made mistakes when solving this, could I be helped in spotting the errors please?

Thanks in advance!

a. Given $\displaystyle f(x)=\sqrt{x-2}+1$, find $\displaystyle f^{\prime}(x)$ using the limit definition of the derivative.

$\displaystyle f^{\prime}(x) = \frac1h \cdot \frac{h+2}{\left(\sqrt{x+h-2}+1+\sqrt{x-2}+1\right)}$

$\displaystyle = \frac{2}{\left(\sqrt{x+h-2}+1+\sqrt{x-2}+1\right)}$

$\displaystyle = \frac{2}{2\sqrt{x-2}+2}$

$\displaystyle = \frac{1}{\sqrt{x-2}+2}$

b. Find the equation of the line tangent to the graph $\displaystyle f(x)=\sqrt{x-2}+1$ at the point x = 6.

$\displaystyle y = f(x)(f^{\prime}(x)(x-x_0)$

$\displaystyle f^{\prime}(x) = \frac{1}{2+2} = \frac14$

$\displaystyle f(x) = \left(\sqrt{6-2}+1\; = 3\right)$

$\displaystyle y = 3\cdot\frac14(x-6)$