Step by step
The question
Question: find the derivative of f(x) = 2 sin x − 4 cos x.
f(x) = 2 sin x − 4 cos x
1 The rules
The derivative of sin x is cos x.
sin x → cos x
The derivative of cos x is −sin x.
Mind the minus!
cos x → −sin x
2 Each term
Use the rule on each term.
A number in front just waits, and multiplies.
f′(x) = 2 × cos x − 4 × (−sin x)
Careful!
The derivative of cos x is minus sin x.
Do not lose the minus.
Careful: not cos x → sin x ✗
Minus times minus is plus.
− × − = +
3 Tidy up
So f′(x) = 2 cos x + 4 sin x.
= 2 cos x + 4 sin x
4 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 4.4465, with a calculator in radians.
f′(1) = 2 cos 1 + 4 sin 1 ≈ 4.4465
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 4.45.
Very close to 4.4465.
Right!
f(1.001) − f(1)0.001 ≈ 4.45
Let's look back.
First: the rules.
Then: each term.
Then: tidy up.
Last: check.
Answer
= 2 cos x + 4 sin x