Natural exponential function and its derivative | Year 12

01 / LEARN

Understand the idea.

Why the base e is special

The natural exponential $e^{x}$ has the special property ${(e^{x})}^{'} = e^{x}$ . Its height equals its tangent gradient, so it is positive and increasing for every real input.

For $e^{g (x)}$ , retain the full exponential and multiply by $g^{'} (x)$ . Changing the exponent changes how quickly the exponential input moves, which is why the chain-rule factor is necessary.

Exponential functions in combinations

Distinguish a variable exponent from a fixed power: $x^{3}$ uses the power rule, while $e^{3 x}$ uses the exponential rule and chain rule. Other changing factors still require the product or quotient rule.

Since $e^{g (x)} > 0$ for real $g (x)$ , an exponential factor cannot make a derivative zero. When finding stationary points, solve the remaining factor and then use the original function for each height.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A scaled exponential

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

Worked solution

1
Identify the exponent and its derivative.
$u = 2 x - 3, u^{'} = 2$
The additive constant in the exponent has derivative zero.
2
Apply the chain rule with the coefficient.
$y^{'} = 4 e^{2 x - 3} \cdot 2$
The exponential stays unchanged and is multiplied by its inner rate.
3
Combine the constant factors.
$y^{'} = 8 e^{2 x - 3}$
Only the coefficient changes; the exponent remains 2x-3.

Answer

$8 e^{2 x - 3}$

EXAMPLE 02

A stationary exponential product

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

Worked solution

1
Differentiate the product.
$\begin{aligned} f^{'} (x) & = 1 \cdot e^{- x} + x (- e^{- x}) \\ = e^{- x} (1 - x) \end{aligned}$
The chain rule gives the negative sign in the derivative of the exponential.
2
Solve for a horizontal tangent.
$e^{- x} > 0, f^{'} (x) = 01 - x = 0 x = 1$
A real exponential is never zero.
3
Find the corresponding height.
$f (1) = 1 e^{- 1} = e^{- 1}$
Substitute into the original curve rather than into its derivative.
4
State and check the stationary point.
$(1, e^{- 1}), f^{'} (x) > 0 (x < 1), f^{'} (x) < 0 (x > 1)$
The sign change also shows that this stationary point is a maximum.

Answer

$(1, e^{- 1})$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

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

For $e^{x^{2}}$ , the outer exponential stays $e^{x^{2}}$ and the inner derivative is $2 x$ . Multiplying gives $2 x e^{x^{2}}$ for every real $x$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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.

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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 ↗

In this lesson

Understand the idea.

Why the base e is special

Exponential functions in combinations

See the method in action.

A scaled exponential

Identify the exponent and its derivative.

Apply the chain rule with the coefficient.

Combine the constant factors.

A stationary exponential product

Differentiate the product.

Solve for a horizontal tangent.

Find the corresponding height.

State and check the stationary point.

Try it yourself.

Check your understanding

Keep practising