From First Principles
Key Questions

Answer:
# d/dx e^x = e^x # Explanation:
We seek:
# d/dx e^x# Method 1  Using the limit definition:
# f'(x) = lim_{h to 0} {f(x+h)f(x)}/{h} # We have:
# f'(x) = lim_{h to 0} {e^(x+h)e^(x)}/{h} #
# " " = lim_{h to 0} {e^xe^he^(x)}/{h} #
# " " = lim_{h to 0} e^x((e^h1))/{h} #
# " " = e^xlim_{h to 0} ((e^h1))/{h} # Think about this limit for a moment and we can rewrite it as:
#lim_{h to 0} ((e^h1))/{h} = lim_{h to 0} ((e^he^0))/{h} #
# " " = lim_{h to 0} ((e^(0+h)e^0))/{h} #
# " " = f'(0) # (by the derivative definition)Hence,
# f'(x) = e^xf'(0) # Now, It can be shown that this limit:
# f'(0) = lim_{h to 0} ((e^h1))/{h} # both exists and is equal to unity. Additionly, the number
#2.718281 ...# , which we call Euler's number) denoted by#e# is extremely important in mathematics, and is in fact an irrational number (like#pi# and#sqrt(2)# ,And so:
# d/dx e^x=e^x# This special exponential function with Euler's number,
#e# , is the only function that remains unchanged when differentiated.Method 2  Power Series
We can use the power series:
# e^x = 1 +x + x^2/(2!) + x^3/(3!) + x^4/(4!) + ... # Then we can differentiate term by term using the power rule:
# d/dx e^x = d/dx{1 +x + x^2/(2!) + x^3/(3!) + x^4/(4!) + ... } # # \ \ \ \ \ \ \ \ \ = 0 +1 + (2x)/(2!) + (3x^2)/(3!) + (4x^3)/(4!) + (5x^4)/(5!) + ... # # \ \ \ \ \ \ \ \ \ = 1 + (x)/(1!) + (3x^2)/(2! * 2) + (4x^3)/(3! * 4) + (5x^4)/(4! * 5) + ... # # \ \ \ \ \ \ \ \ \ = 1 +x + x^2/(2!) + x^3/(3!) + x^4/(4!) + ... # 
By rewriting a bit,
#y=c^x=e^{(lnc)x}# .By Chain Rule,
#y'=e^{(lnc)x}cdot(lnc)=(lnc)c^x#
I hope that this was helpful.