Quotient or Power Rule?

J

Jason76

Senior Member
Joined
Oct 19, 2012
Messages
1,180
\(\displaystyle f(x) = \dfrac{t - \sqrt{t}}{t^{1/7}}\)

I would go with power rule, but first the expression has to be changed to

\(\displaystyle f(x) = \dfrac{t}{t^{1/7}} - \dfrac{\sqrt{t}}{t^{1/7}}\)

\(\displaystyle f(x) = \dfrac{t}{t^{1/7}} - \dfrac{t^{1/2}}{t^{1/7}}\)

Next differentiate:
 
looks good if f(t) is what you want.
To differentiate, quotient, power and chain rules apply.
t > 0
 
Last edited:
Jason76 said:
\(\displaystyle f(x) = \dfrac{t - \sqrt{t}}{t^{1/7}}\)

I would go with power rule, but first the expression has to be changed to

\(\displaystyle f(x) = \dfrac{t}{t^{1/7}} - \dfrac{\sqrt{t}}{t^{1/7}}\)

\(\displaystyle f(x) = \dfrac{t}{t^{1/7}} - \dfrac{t^{1/2}}{t^{1/7}}\)

Next differentiate:
Click to expand...
You need to do a little more before differentiating:
\(\displaystyle \dfrac{t}{t^{1/7}}= t^{1- 1/7}= t^{6/7}\) and
\(\displaystyle \dfrac{t^{1/2}}{t^{1/7}}= t^{1/2-1/7}= t^{7/14- 2/14}= t^{5/14}\)
 
Jason76 said:
\(\displaystyle f(x) = \dfrac{t - \sqrt{t}}{t^{1/7}}\)

I would go with power rule, but first the expression has to be changed to

\(\displaystyle f(x) = \dfrac{t}{t^{1/7}} - \dfrac{\sqrt{t}}{t^{1/7}}\)

\(\displaystyle f(x) = \dfrac{t}{t^{1/7}} - \dfrac{t^{1/2}}{t^{1/7}}\)

Next differentiate:
Click to expand...

Differentiate with respect to what?

Is 't' a function of 'x'?
 
Jason76 said:
\(\displaystyle f(x) = \dfrac{t - \sqrt{t}}{t^{1/7}}\)
Click to expand...
Jason76, pay attention. This is at least the second time that I have

seen this type of post. Evidently, there is supposed to be the same variable

throughout the problem, either x or t. Please go back and recheck the problem.
 
Let's try with quotient rule:

\(\displaystyle \dfrac{t - t^{1/2}}{t}\)

\(\displaystyle \dfrac{t(1 - \dfrac{t}{2}^{-1/2}) - (t - t^{1/2})(1)}{1^{2}}\)

\(\displaystyle \dfrac{t(1 - \dfrac{t}{2}^{-1/2}) - (t - t^{1/2})(1)}{1}\)

\(\displaystyle \dfrac{(t - \dfrac{t}{2}^{1/2}) - (t - t^{1/2})}{1}\)

\(\displaystyle \dfrac{t - \dfrac{t}{2}^{1/2} - t + t^{1/2}}{1}\) :confused:
 
Last edited:
Jason76 said:
\(\displaystyle f(x) = \dfrac{t - \sqrt{t}}{t^{1/7}}\)

I would go with power rule, but first the expression has to be changed to

\(\displaystyle f(x) = \dfrac{t}{t^{1/7}} - \dfrac{\sqrt{t}}{t^{1/7}}\)

\(\displaystyle f(x) = \dfrac{t}{t^{1/7}} - \dfrac{t^{1/2}}{t^{1/7}}\)

Next differentiate:
Click to expand...

\(\displaystyle f(x) = t^{6/7} - t^{5/14}\)

\(\displaystyle \displaystyle \frac{d}{dx}[f(x)] \ = \ f'(x) \ = \frac{6}{7}t^{-\frac{1}{7}} * \frac{dt}{dx} \ - \ \frac{5}{14}t^{-\frac{9}{14}} * \frac{dt}{dx}\)
 
You must log in or register to reply here.
Top