Step by step
The question
Question: find the derivative of f(x) = 3 cos x.
f(x) = 3 cos x
1 The rules
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) = 3 × (−sin x)
Careful!
The derivative of cos x is minus sin x.
Do not lose the minus.
Careful: not cos x → sin x ✗
3 Tidy up
So f′(x) = −3 sin x.
= −3 sin x
4 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives −2.5244, with a calculator in radians.
f′(1) = −3 sin 1 ≈ −2.5244
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about −2.53.
Very close to −2.5244.
Right!
f(1.001) − f(1)0.001 ≈ −2.53
Let's look back.
First: the rules.
Then: each term.
Then: tidy up.
Last: check.
Answer
= −3 sin x