Posted: Feb 18, 2012 10:56 am
I will not have time to go through all the rest of the examples today; so I will just go through the second equation and solve this one in this installment.
The above is solved by seeing it as the sum of two terms; each term will be differentiated and their sum will be the first derivative of the above function. The same is true for the second derivative.
So we can say that:
and the derivative will be:
Let's start with [math]. In this case we have a=5, n=7, and n-1=7-1=6. So, the derivative will be:
This one was easy, if you follow the rules about differentiation I gave earlier. How about the second term? It is a negative power. Does that play any role in differentiating? The answer is no. Whether n is positive or negative it does not play any role at all in differentiating powers.
So for [math], we have a=1, n=-5, and n-1=-5-1=-6. Be careful with the signs here, as it is easy to make a mistake, and change the result. So, our result here is:
Putting the two derivatives together, we get:
And that's your result, for the first derivative.
The problem asks for the second derivative as well and here we have as terms:
So what is the second derivative of the first term? For [math] we have a=35, n=6, n-1=6-1=5. Then this derivative will be:
What about the second term? Here we have a=-5, n=-6 and n-1=-6-1=7. Plugging in the numbers we have:
And the resulting second derivative will be:
Tomorrow, I will try and solve the other two examples.
[math]
The above is solved by seeing it as the sum of two terms; each term will be differentiated and their sum will be the first derivative of the above function. The same is true for the second derivative.
So we can say that:
[math] and
[math] then:
[math]
and the derivative will be:
[math]
Let's start with [math]. In this case we have a=5, n=7, and n-1=7-1=6. So, the derivative will be:
[math]
[math]
This one was easy, if you follow the rules about differentiation I gave earlier. How about the second term? It is a negative power. Does that play any role in differentiating? The answer is no. Whether n is positive or negative it does not play any role at all in differentiating powers.
So for [math], we have a=1, n=-5, and n-1=-5-1=-6. Be careful with the signs here, as it is easy to make a mistake, and change the result. So, our result here is:
[math]
Putting the two derivatives together, we get:
[math]
And that's your result, for the first derivative.
The problem asks for the second derivative as well and here we have as terms:
[math]
[math]
So what is the second derivative of the first term? For [math] we have a=35, n=6, n-1=6-1=5. Then this derivative will be:
[math]
[math]
What about the second term? Here we have a=-5, n=-6 and n-1=-6-1=7. Plugging in the numbers we have:
[math]
[math]
And the resulting second derivative will be:
[math]
Tomorrow, I will try and solve the other two examples.