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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 