The derivative of f(x) = b^{ x} is given by
f '(x) = b^{ x} ln b
Note: if f(x) = e^{ x} , then f '(x) = e^{ x}
Example 1: Find the derivative of f(x) = 2^{ x}
Solution to Example 1:
 Apply the formula above to obtain
f '(x) = 2^{ x} ln 2
Example 2: Find the derivative of f(x) = 3^{ x} + 3x^{ 2}
Solution to Example 2:
 Let g(x) = 3^{ x} and h(x) = 3x^{ 2}, function f is the sum of functions g and h: f(x) = g(x) + h(x). Use the sum rule, f '(x) = g '(x) + h '(x), to find the derivative of function f
f '(x) = 3^{ x} ln 3 + 6x
Example 3: Find the derivative of f(x) = e^{ x} / ( 1 + x )
Solution to Example 3:
 Let g(x) = e^{ x} and h(x) = 1 + x, function f is the quotient of functions g and h: f(x) = g(x) / h(x). Hence we use the quotient rule, f '(x) = [ h(x) g '(x)  g(x) h '(x) ] / h(x)^{ 2}, to find the derivative of function f.
g '(x) = e^{ x}
h '(x) = 1
f '(x) = [ h(x) g '(x)  g(x) h '(x) ] / h(x)^{ 2}
= [ (1 + x)(e^{ x})  (e^{ x})(1) ] / (1 + x)^{ 2}
 Multiply factors in the numerator and simplify
f '(x) = x e^{ x} / (1 + x)^{ 2}
Example 4: Find the derivative of f(x) = e^{ 2x + 1}
Solution to Example 4:
 Let u = 2x + 1 and y = e^{ u}, Use the chain rule to find the derivative of function f as follows.
f '(x) = (dy / du) (du / dx)
 dy / du = e^{ u} and du / dx = 2
f '(x) = (e^{ u})(2) = 2 e^{ u}
 Substitute u = 2x + 1 in f '(x) above
f '(x) = 2 e^{ 2x + 1}
Exercises Find the derivative of each function.
1  f(x) = e^{ x} 2^{ x}
2  g(x) = 3^{ x}  3x^{ 3}
3  h(x) = e^{ x} / (2x  3)
4  j(x) = e^{ (x2 + 2)}
solutions to the above exercises
1  f '(x) = e^{ x} 2^{ x} ( ln 2 + 1)
2  g '(x) = 3^{ x} ln 3  9x^{ 2}
3  h '(x) = e^{ x}(2x 5) / (2x  3)^{ 2}
4  j '(x) = 2x e^{ (x2 + 2)}
More on differentiation and derivatives
