First Derivative of Exponential Functions to any Base

Example 1

Solution to Example 1

• Apply the formula above to obtain
  f '(x) = 2 ^x ln 2

Example 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

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

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

Solutions to the Above Exercises

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