Natural exponential function and its derivative | Year 12

01 / LEARN

The idea, step by step

Why the base e is special

The function $e^{x}$ has a derivative equal to itself. Its tangent gradient at any point is therefore the height $e^{x}$ , and is always positive.

For a composite exponent, the chain rule remains essential: the derivative of $e^{g (x)}$ is $g^{'} (x) e^{g (x)}$ .

Exponential functions in combinations

Keep the exponential factor when differentiating; it does not lose a power as a polynomial does. Use product or quotient rules if other changing factors are present.

An exponential factor is never zero for a real exponent. In a stationary-point equation this can reduce the problem to the other factor.

A common trap

Treating $e^{2 x}$ as though the power rule changes it to $2 e^{x}$ .

Instead: For a variable exponent, retain $e^{2 x}$ and multiply by the exponent derivative 2.

02 / WORKED EXAMPLES

See the thinking

Example 1

A scaled exponential

Differentiate $y = 4 e^{2 x - 3}$ .

The derivative of the exponent $2 x - 3$ is 2.
Multiply the unchanged exponential by $4 \times 2$ .

8 e^{2 x - 3}

Example 2

A stationary exponential product

Find the stationary point of $f (x) = x e^{- x}$ .

$f^{'} (x) = e^{- x} (1 - x)$ .
Since $e^{- x} > 0$ , the derivative is zero only at $x = 1$ .
$f (1) = e^{- 1}$ .

(1, e^{- 1})

Check your understanding

Differentiate $e^{x^{2}}$ .

Show answer

$2 x e^{x^{2}}$

Use the chain rule; the exponent derivative is $2 x$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Differentiate $7 e^{x} - 3$ .
2 marks
Differentiate $e^{- 3 x}$ .
2 marks
Differentiate $e^{x^{2} + 2 x}$ .
2 marks
Find the tangent to $y = e^{2 x}$ at $x = 0$ .
2 marks
Differentiate $x^{2} e^{- x}$ .
3 marks
Explain why $f (x) = e^{x^{2}}$ decreases for $x < 0$ although $e^{u}$ increases with $u$ .
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Differentiate $7 e^{x} - 3$ .
2 marks
Use ${(e^{x})}^{'} = e^{x}$ .
The constant differentiates to zero.
$7 e^{x}$
Differentiate $e^{- 3 x}$ .
2 marks
Retain the exponential.
Multiply by the exponent derivative $- 3$ .
$- 3 e^{- 3 x}$
Differentiate $e^{x^{2} + 2 x}$ .
2 marks
Differentiate the exponent.
Multiply by the unchanged exponential.
$(2 x + 2) e^{x^{2} + 2 x}$
Find the tangent to $y = e^{2 x}$ at $x = 0$ .
2 marks
The point is $(0, 1)$ .
The derivative is $2 e^{2 x}$ , giving gradient 2.
$y = 2 x + 1$
Differentiate $x^{2} e^{- x}$ .
3 marks
Use the product rule.
The derivative of $e^{- x}$ is $- e^{- x}$ .
$e^{- x} (2 x - x^{2})$
Explain why $f (x) = e^{x^{2}}$ decreases for $x < 0$ although $e^{u}$ increases with $u$ .
3 marks
The derivative is $2 x e^{x^{2}}$ .
The exponential is positive but $2 x < 0$ on $x < 0$ .
Its inner input $x^{2}$ decreases as $x$ increases towards zero.

Checking saved progress…

Continue learning

Exponential growth and decay

Next lesson

Curriculum reference & lesson scope

SACE Stage 2 Mathematical Methods

Further differentiation and applications from the current SACE Methods outline. Logarithmic derivatives have their own chapter; integration follows later. Trigonometric calculus uses radians.

Official curriculum source ↗

Back to Differentiation and applications