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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
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