Worked examples
The question
Question: find the derivative of f(x) = 2ˣ.
f(x) = 2ˣ
1 Write it with e
We know how to take the derivative of e to a power.
And any number is e to a power: 2 is e to the ln 2.
2 = e^(ln 2)
So 2 to the x is e to the x times ln 2.
2ˣ = e^(x ln 2)
2 The rule
The derivative of e to a power is itself, times the slope of the power: ln 2.
ddxe^(x ln 2) = e^(x ln 2) × ln 2
The rule: b to the x gives b to the x, times ln b.
bˣ → bˣ ln b
Careful!
The power rule is only for x to a fixed number.
Here x is in the power: use b to the x, times ln b.
Careful: not 2ˣ → x2^(x − 1) ✗
3 Tidy up
Put it back in base 2: f′(x) = 2ˣ ln 2.
f′(x) = 2ˣ ln 2
4 Check
Let's check: the derivative is the slope.
At x = 1, our answer gives 1.3863, with a calculator.
f′(1) = 1.3863
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 1.39.
Very close to 1.3863.
Right!
f(1.001) − f(1)0.001 ≈ 1.39
Let's look back.
First: write it with e.
Then: the rule.
Then: tidy up.
Last: check.
▶ Watch it solved
The question
Question: find the derivative of f(x) = 3^(2x).
f(x) = 3^(2x)
1 Write it with e
We know how to take the derivative of e to a power.
And any number is e to a power: 3 is e to the ln 3.
3 = e^(ln 3)
So 3 to the 2x is e to the 2x times ln 3.
3^(2x) = e^((2x) ln 3)
2 The rule
The derivative of e to a power is itself, times the slope of the power: 2 times ln 3.
ddxe^((2x) ln 3) = e^((2x) ln 3) × 2 × ln 3
The rule: b to the x gives b to the x, times ln b.
bˣ → bˣ ln b
Careful!
The power rule is only for x to a fixed number.
Here x is in the power: use b to the x, times ln b.
Careful: not 3ˣ → x3^(x − 1) ✗
3 Tidy up
Put it back in base 3, with the numbers in front: f′(x) = 2 × 3^(2x) ln 3.
f′(x) = 2 × 3^(2x) ln 3
4 Check
Let's check: the derivative is the slope.
At x = 1, our answer gives 19.775, with a calculator.
f′(1) = 19.775
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 19.8.
Very close to 19.775.
Right!
f(1.001) − f(1)0.001 ≈ 19.8
Let's look back.
First: write it with e.
Then: the rule.
Then: tidy up.
Last: check.
Answer f′(x) = 2 × 3^(2x) ln 3
▶ Watch it solved
The question
Question: find the derivative of f(x) = 2^(x²).
f(x) = 2^(x²)
1 Write it with e
We know how to take the derivative of e to a power.
And any number is e to a power: 2 is e to the ln 2.
2 = e^(ln 2)
So 2 to the x² is e to the x² times ln 2.
2^(x²) = e^((x²) ln 2)
2 The rule
The derivative of e to a power is itself, times the slope of the power: 2x times ln 2.
ddxe^((x²) ln 2) = e^((x²) ln 2) × 2x × ln 2
The rule: b to the x gives b to the x, times ln b.
bˣ → bˣ ln b
Careful!
The power rule is only for x to a fixed number.
Here x is in the power: use b to the x, times ln b.
Careful: not 2ˣ → x2^(x − 1) ✗
3 Tidy up
Put it back in base 2, with the numbers in front: f′(x) = 2x × 2^(x²) ln 2.
f′(x) = 2x × 2^(x²) ln 2
4 Check
Let's check: the derivative is the slope.
At x = 1, our answer gives 2.7726, with a calculator.
f′(1) = 2.7726
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 2.78.
Very close to 2.7726.
Right!
f(1.001) − f(1)0.001 ≈ 2.78
Let's look back.
First: write it with e.
Then: the rule.
Then: tidy up.
Last: check.
Answer f′(x) = 2x × 2^(x²) ln 2
▶ Watch it solved
