Exponential and logarithmic functions and their derivatives
y = eˣ and y = log x are inverse functions. Their derivatives are simple: d/dx eˣ = eˣ and d/dx log x = 1/x for x > 0.
The two functions
An exponential function is y = bˣ with base b > 1. Its domain is all real numbers, its range is y > 0, and it passes through (0, 1). The most important base is e ≈ 2.718. The logarithmic function y = log_b x is its inverse: log_b x = y exactly when bʸ = x. Its domain is x > 0, so the log of 0 or of a negative number is not defined. Here log x means logₑ x, the natural logarithm. We write a^(b) for a raised to the power b when the power is long.
eˣ and log x side by side
| Property | y = eˣ | y = log x |
|---|---|---|
| Domain | All real numbers | x > 0 |
| Range | y > 0 | All real numbers |
| Passes through | (0, 1) | (1, 0) |
| Shape | Always increasing; hugs the negative x-axis | Always increasing; hugs the negative y-axis near x = 0 |
| Derivative | eˣ | 1/x |
How to draw the graphs of y = eˣ and y = log x
- Draw the x and y axes, and the line y = x as a dashed line.
- For y = eˣ, plot (−1, 1/e), (0, 1) and (1, e), with e ≈ 2.7. Draw a rising curve that comes closer and closer to the negative x-axis but never meets it.
- For y = log x, plot (1/e, −1), (1, 0) and (e, 1). Draw a rising curve that runs down along the negative y-axis, close to it but never meeting it.
- The two curves are mirror images in the line y = x. Label each curve and the points (0, 1) and (1, 0).
Derivatives with the chain rule
| Function | Derivative |
|---|---|
| eˣ | eˣ |
| e^(f(x)) | e^(f(x)) × f′(x) |
| log x, x > 0 | 1/x |
| log f(x), f(x) > 0 | f′(x)/f(x) |
| aˣ, a > 0 | aˣ log a |
Example 1: direct use
Differentiate (i) e^(x²), (ii) log(sin x), 0 < x < π, (iii) 5ˣ.
- (i) Chain rule: d/dx e^(x²) = e^(x²) × 2x = 2x e^(x²)
- (ii) d/dx log(sin x) = (1/sin x) × cos x = cot x
- (iii) d/dx 5ˣ = 5ˣ log 5
(i) 2x e^(x²); (ii) cot x; (iii) 5ˣ log 5
Example 2: board style
Differentiate y = log(x + √(x² + 1)).
- Let u = x + √(x² + 1). Then dy/dx = (1/u) × du/dx.
- du/dx = 1 + 2x/(2√(x² + 1)) = 1 + x/√(x² + 1)
- = (√(x² + 1) + x)/√(x² + 1)
- dy/dx = [1/(x + √(x² + 1))] × [(x + √(x² + 1))/√(x² + 1)]
- The factor x + √(x² + 1) cancels.
dy/dx = 1/√(x² + 1)
Bacteria in idli batter and curd multiply roughly exponentially while they ferment. Sound levels in decibels and pH in chemistry use logarithms to squeeze huge ranges into small numbers.
1 marks: “Find d/dx (e^(2x + 3)).” Chain rule: e^(2x + 3) × 2 = 2e^(2x + 3).
2 marks: “Differentiate y = e^(tan x) + log(sec x), 0 < x < π/2.” dy/dx = sec²x e^(tan x) + (sec x tan x)/sec x = sec²x e^(tan x) + tan x.
2 marks: “Differentiate y = sin(log x), x > 0.” dy/dx = cos(log x) × (1/x) = cos(log x)/x.
eˣ stays eˣ; log x becomes 1/x.