Another dy/dx radical problem with no rules allowed

[QuarkCharmer]

Homework Statement

$$f(t) = t^2- \sqrt{t}$$

Find f'(t)

Homework Equations

Difference Quotient

The Attempt at a Solution

$$f(t) = t^2-\sqrt{t}$$

$$\lim_{h\to0} \frac{(t+h)^2-\sqrt{t+h}-t^2+\sqrt{t}}{h}$$

$$\lim_{h\to0} \frac{t^2+2th+h^2-\sqrt{t+h}-t^2+\sqrt{t}}{h}$$

$$\lim_{h\to0} \frac{2th+h^2-\sqrt{t+h}+\sqrt{t}}{h}$$

Here is where I get lost. I am trying to do it this way, but I am open to any suggestions.

$$\lim_{h\to0} 2t+h + \frac{-\sqrt{t+h}+\sqrt{t}}{h}$$

$$\lim_{h\to0} 2t+h + \frac{(-\sqrt{t+h}+\sqrt{t})(-\sqrt{t+h}-\sqrt{t})}{h(-\sqrt{t+h}-\sqrt{t})}$$

$$\lim_{h\to0} 2t+h + \frac{-\sqrt{t+h}^2 - \sqrt{t}^2}{h(-\sqrt{t+h}-\sqrt{t})}$$

$$\lim_{h\to0} 2t+h + \frac{-t-h-t}{h(-\sqrt{t+h}-\sqrt{t})}$$

$$\lim_{h\to0} 2t+h + \frac{-2t-h}{h(-\sqrt{t+h}-\sqrt{t})}$$

Now what? The more I expand and contract it, the more complicated it becomes with no hopes of getting that h out of the denominator.

 

[lanedance]

for a start, i would do each part sperately, rather than dealing with a jumble of terms ie.
$$\frac{d}{dt}f(t) = \frac{d}{dt}(t^2-\sqrt{t}) =\lim_{h\to0} \frac{(t+h)^2-\sqrt{t+h}-t^2+\sqrt{t}}{h}$$

$$=(\lim_{h\to0} \frac{(t+h)^2-t^2}{h}) - (\lim_{h\to0} \frac{\sqrt{t+h}-\sqrt{t}}{h})$$

we did the sqrt in the last problem (myultiply by conjugate), and the t^2 term should simplify reasonably easy

 

[Char. Limit]

I think your problem is that you set $(-\sqrt{t+h} + \sqrt{t}) (-\sqrt{t+h} - \sqrt{t}) = -(t+h) - t$, when it should be (t+h) - t.

 

[QuarkCharmer]

I think your problem is that you set $(-\sqrt{t+h} + \sqrt{t}) (-\sqrt{t+h} - \sqrt{t}) = -(t+h) - t$, when it should be (t+h) - t.

I think I distributed that correctly?

$$\lim_{h\to0} 2t+h + \frac{-\sqrt{t+h}^2 - \sqrt{t}^2}{h(-\sqrt{t+h}-\sqrt{t})}$$

$$\lim_{h\to0} 2t+h + \frac{-t-h-t}{h(-\sqrt{t+h}-\sqrt{t})}$$

-(root(t+h)^2)-root(t)^2
-(t+h)-t
-t-h-t
right?

 

[Char. Limit]

Not quite. Remember that you have

$$\left(-\sqrt{t+h}\right)^2$$

not

$$-\left(\sqrt{t+h}\right)^2$$

You see?

 

[QuarkCharmer]

Yeah I understand the mistake there. I think I just made it so complicated that it was easy to make silly mistakes.

I used lanedance's suggestion. Worked great. I'll have to remember to check to see if the numerator can be arranged and a negative 1 factored out to put the limits together like that.

Thanks again.

Solution was f'(x) = 2t - 1/(2root(t))

 

[Char. Limit]

I certainly hope you mean f'(t), but other than that good.

 

[QuarkCharmer]

I certainly hope you mean f'(t), but other than that good.

lol, yes I do. I just did the paperwork where t = x, because my + looks like t.

Appreciate the help.

 

[lanedance]

so if
f(t) = g(t) + h(t)
you can show
f'(t) = g'(t) + h'(t)
which is what you did above

with a lot of these you should know what you're expecting to happen, so you can leverage off that to help in simplfying the limits

 

[Mark44]

so if
f(t) = g(t) + h(t)
you can show
f'(t) = g'(t) + h'(t)
which is what you did above

with a lot of these you should know what you're expecting to happen, so you can leverage off that to help in simplfying the limits

lanedance,
Based on the title of this thread, which includes "... no rules allowed," I think that QuarkCharmer is expected to find the derivative without using the sum rule or other rules for derivatives. At least that's where I think you're going with this advice.

 

[lanedance]

not quite - i was still implying to show everything form first principles. However the sum rule falls out quite quickly and simplifies the working early.

Even though we can't use those predefined formulas explicitly, i find i still usually either prove them or use them as a hint for which direction to head in.

 

[Char. Limit]

Are you allowed to prove the sum rule from first principles, and use it in the problem, maybe?

 

[QuarkCharmer]

I thought about proving some rules for these types of things. Something along the lines of working out the derivative of x^n using the difference quotient, (Which would radically speed up my time deriving polynomials and the likes), but I really think that I am expected to work it all out in one giant pile of algebra :(

All the problems seem "doable" so far without using rules of any kind, I have it whittled down to 3 more that must require some algebraic finesse to complete, but I haven't given up on those 3 yet as to post them here.

Again, I appreciate all the help. PF is the best!