Worked examples
The question
Question: find the derivative of f(x) = sin⁻¹ x.
f(x) = sin⁻¹ x
1 The rule
y = sin⁻¹ x means: sin y = x.
y is the angle whose sine is x.
y = sin⁻¹ x means sin y = x
Take the derivative of both sides: cos y times y′ is 1.
So y′ is 1 over cos y.
And cos y is the square root of 1 minus sin² y, which is 1 minus x².
cos y × y′ = 1
y′ = 1cos y = 1√1 − x²
The rule: the derivative of sin⁻¹ x is 1 over √(1 − x²).
sin⁻¹ x → 1√1 − x²
Careful!
The little −1 means the inverse function, the angle.
It does not mean 1 over sin x.
Careful: sin⁻¹ x is not 1sin x
2 Use the rule
So f′(x) = 1 over √(1 − x²).
f′(x) = 1√1 − x²
3 Check
Let's check: the derivative is the slope.
At x = 0.5, our answer gives 1.1547, with a calculator in radians.
f′(0.5) = 1.1547
Now the slope between two very close points: x = 0.5 and x = 0.501.
A calculator gives about 1.16.
Very close to 1.1547.
Right!
f(0.501) − f(0.5)0.001 ≈ 1.16
Let's look back.
First: the rule.
Then: use the rule.
Last: check.
▶ Watch it solved
The question
Question: find the derivative of f(x) = tan⁻¹(2x).
f(x) = tan⁻¹(2x)
1 The rule
The rule: the derivative of tan⁻¹ u is 1/(1 + u²).
tan⁻¹ u → 11 + u²
2 The inside
There is something inside: 2x.
Call it u.
u = 2x
3 The outside
First the outside: the rule on tan⁻¹ u.
dydu = 11 + u²
4 The inside's slope
The inside's slope: the power rule on 2x gives 2.
dudx = 2
5 Multiply
The chain rule: multiply the two, and put 2x back in place of u.
f′(x) = 11 + (2x)2 × 2
Careful!
Do not stop at the outside.
Multiply by the inside's slope, 2.
Careful: not 11 + (2x)2 ✗
Square the inside: (2x)² is 4x².
(2x)2 = 4x2
Tidy up: f′(x) = 2/(1 + 4x²).
= 21 + 4x2
6 Check
Let's check: the derivative is the slope.
At x = 1, our answer gives 0.4, with a calculator in radians.
f′(1) = 0.4
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 0.4.
Very close to 0.4.
Right!
f(1.001) − f(1)0.001 ≈ 0.4
Let's look back.
First: the rule.
Then: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
▶ Watch it solved
The question
Question: find the derivative of f(x) = sin⁻¹(x²).
f(x) = sin⁻¹(x2)
1 The rule
The rule: the derivative of sin⁻¹ u is 1 over √(1 − u²).
sin⁻¹ u → 1√1 − u²
2 The inside
There is something inside: x².
Call it u.
u = x2
3 The outside
First the outside: the rule on sin⁻¹ u.
dydu = 1√1 − u²
4 The inside's slope
The inside's slope: the power rule on x² gives 2x.
dudx = 2x
5 Multiply
The chain rule: multiply the two, and put x² back in place of u.
f′(x) = 1√1 − (x2)2 × 2x
Careful!
Do not stop at the outside.
Multiply by the inside's slope, 2x.
Careful: not 1√1 − (x2)2 ✗
Square the inside: (x²)² is x⁴.
(x2)2 = x4
Tidy up: f′(x) = 2x over √(1 − x⁴).
= 2x√1 − x4
6 Check
Let's check: the derivative is the slope.
At x = 0.5, our answer gives 1.0328, with a calculator in radians.
f′(0.5) = 1.0328
Now the slope between two very close points: x = 0.5 and x = 0.501.
A calculator gives about 1.03.
Very close to 1.0328.
Right!
f(0.501) − f(0.5)0.001 ≈ 1.03
Let's look back.
First: the rule.
Then: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
▶ Watch it solved
