Question

I'm having a bit of trouble finding the derivative of g(t) = 7/sqrt(t) 'algebraically'. What steps should I take to solve this without using the power rule?

Answer 1

you should use the power rule, but do you know the quotient rule?

Answer 2

Without using the power rule?
What do you mean?

Like using the limit definition of the derivative?

Oh also, \(\rm \color{forestgreen}{\text{Welcome to OpenStudy! :)}}\)

Answer 3

Yes, by using the limit definition! Thanks :)

Answer 4

\[\large\rm g(\color{orangered}{t})=\frac{7}{\sqrt{\color{orangered}{t}}}\]Evaluating our function at t+h instead of t,\[\large\rm g(\color{orangered}{t+h})=\frac{7}{\sqrt{\color{orangered}{t+h}}}\]So these are the two pieces we'll need to use for our difference quotient.

Answer 5

\[\large\rm g'(t)=\lim_{h\to0}\frac{g(t+h)-g(t)}{h}\]and so,\[\large\rm =\lim_{h\to0}\frac{\dfrac{7}{\sqrt {t+h}}-\dfrac{7}{\sqrt t}}{h}\]

Answer 6

We'll need to apply some fancy Algebra tricks to simplify this a bit.

Answer 7

If we multiply the numerator and denominator by the LCM of the denominators,\[\large\rm =\lim_{h\to0}\frac{\dfrac{7}{\sqrt {t+h}}-\dfrac{7}{\sqrt t}}{h}\color{royalblue}{\left(\frac{\sqrt{t+h}\sqrt{t}}{\sqrt{t+h}\sqrt{t}}\right)}\]we can git rid of these fractions upon fractions.

Answer 8

If that's too confusing, you can instead look for a common denominator. I think that will end up being more work though.

Answer 9

That makes sense. I just can't seem to get the denominator simplified enough to get something other than a zero. Thanks for your help!

Answer 10

You'll have to keep shuffling things around until you're able to cancel out that h from the denominator.

Answer 11

Multiplying through by that blue thing gives us,\[\large\rm \lim_{h\to0}\frac{7\sqrt t-7\sqrt{t+h}}{h \sqrt{t+h}\sqrt t}\]Err let's actually leave the 7 outside, should make things easier.\[\large\rm 7\lim_{h\to0}\frac{\sqrt t-\sqrt{t+h}}{h \sqrt{t+h}\sqrt t}\]Multiply through by the `conjugate` of the numerator,\[\large\rm 7\lim_{h\to0}\frac{\sqrt t-\sqrt{t+h}}{h \sqrt{t+h}\sqrt t}\color{forestgreen}{\left(\frac{\sqrt{t}+\sqrt{t+h}}{\sqrt{t}+\sqrt{t+h}}\right)}\]Don't distribute in the denominator though!
Just leave this mess like it is.

Answer 12

Multiplying by conjugates gives us this nice numerator,\[\large\rm 7\lim_{h\to0}\frac{t-(t+h)}{h \sqrt{t+h}\sqrt t(\sqrt{t}+\sqrt{t+h})}\]Should be able to get somewhere from this point, ya? :d

Answer 13

Ahh perfect, I'm dumb lol. Thanks a ton!