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# Find the derivative of esin x

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 2x2

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 2x

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 e1/x

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 ex1/2

Find the derivative of the function:

We apply the formula for the derivative of and the chain rule,

# Find the derivative of e-x2

Find the derivative of the function:

We apply the formula for the derivative of and the chain rule,

# Find the derivative of e4x2

Find the derivative of the function:

We apply the formula for the derivative of and the chain rule,

# Find the derivative of e3x-1

Find the derivative of the function:

We apply the formula for the derivative of and the chain rule,

# Find an approximation of log10 e

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

# Prove some inequalities about (1+1/n)^n for positive integers n

1. For prove,

2. For , prove

1. Proof. Recall the Binomial theorem,

So, letting and we have,

2. Proof. First, we prove the left inequality,

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