# Equation for tangent line for $f(x) = 1/\sqrt{x}$ at $x=a$

first time math stack-exchange-er here.

I'm self-teaching single variable calculus using MIT's free online courses and I think I found a typo in the homework solution set (problem 1C-4 part d). I'm not confident enough in my own abilities to know for sure if this is a mistake vs. my poor math skills.

Could anyone tell me if the following equation is correct?

The problem is as follows: Write an equation for the tangent line for the following functions:

$$f(x) = \frac1{\sqrt{x}}\ \text{ at } x=a$$

I did the following. First I found the derivative of $f(x)$: $$f'(x) = -\frac12 x^{-\frac32}\$$

Then I plugged in a to get $f(a)$ and $f'(a)$ and used the point-slope method to find the equation for the tangent line: $$y-a^{-\frac12} = -\frac12 a^{-\frac32}(x-a)\$$

Which I then simplified to: $$y=-\frac12a^{-\frac32}x + \frac32 a^{-\frac12}\$$

However, the solution set says the answer is: $$y=-a^{-\frac32}x + \frac32 a^{-\frac12}\$$

Is the solution set correct? If so, where did the $-\frac12$ go?

Thank you!

• Welcome to Math.SE! A very excellently asked first question. – Sean Roberson Jul 6 '17 at 21:18
• I get the same answer as you did, seems solution set answer is wrong. – coffeemath Jul 6 '17 at 21:19
• You are right ... there is a mistake in their solutions. – Donald Splutterwit Jul 6 '17 at 21:21
• Don’t forget: the answers to problems are often supplied by underpaid and overworked graduate students. And it would be unheard of for them to be checked by the author(s) of the text. – Lubin Jul 7 '17 at 0:09
• Also, notice another typo in it: the minus sign in $a^{3/2}$ is omitted. – farruhota Jul 7 '17 at 14:02