Find the derivative of the following function:
We apply the chain rule and the formula for the derivative of the exponential to compute,
Find the derivative of the following function:
We apply the chain rule and the formula for the derivative of the exponential to compute,
Find the derivative of the following function:
First, we use the definition of the exponential to rewrite this as,
Then, we can take the derivative of this,
Find the derivative of the function:
First, using the definition of the exponential we rewrite in a form we know what to do with,
Then, we can take the derivative using the chain rule and the formula for the derivative of an exponential,
Find the derivative of the function:
Using the formula for the derivative of the exponential and the chain rule we compute,
Find the derivative of the function:
We apply the formula for the derivative of and the chain rule,
Find the derivative of the function:
We apply the formula for the derivative of and the chain rule,
Find the derivative of the function:
We apply the formula for the derivative of and the chain rule,
Find the derivative of the function:
We apply the formula for the derivative of and the chain rule,
Using the fact that
compute using the solution of the previous exercise (Section 6.9, Exercise #32 of Apostol).
Since we know from the solution of the previous exercise that
we let , , to get
So, letting and we have,
If , then , so using the Binomial theorem we have,
Where we know the inequality is strict since there is at least one term (which is necessarily positive) in since .
Next, we prove the middle inequality,
By part (a) we know,
Further, for we have for all . Thus, we know that,
Hence,
for all . Therefore, we have established the second inequality,
Finally, we prove the right inequality,
Here we expand the first few terms and use a previous result,
for all . In the second to last line we used this result on the th powers of a real number (in this case ). This completes the proof for all of the inequalities requested