Using the definition of limit to find the derivative of $x \mapsto \sqrt{5x}$

Question

Use the definition of limit to find the derivative of $x \mapsto \sqrt{5x}$.

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Answer key:

(sqrt(5)) / (2(sqrt(x)))

What I have tried:

Why won't I get the right answer? What am I doing wrong?

Answer 1

Your answer is the same as the one provided by the answer key. Observe:

$\frac{5}{2\sqrt{5x}}=\frac{5}{2\sqrt{5}\sqrt{x}}$

Multiplying by $\frac{\sqrt{5}}{\sqrt{5}}$ yields $\frac{5\sqrt{5}}{10\sqrt{x}}=\frac{\sqrt{5}}{2\sqrt{x}}$ Which is what the answer key provided.

It is strange that the answer key would be in that form given the vast majority of people would have used a process almost identical to yours, getting the same answer, and that you would get the same form if you used the product rule and chain rule. Although it's always a good idea to manipulate your answer if you are confident in your method, especially if it is a multiple-choice test in which your form does not appear.