Step by step
The question
Question: find the derivative of f(x) = x³ + 2 sin x.
f(x) = x3 + 2 sin x
1 The rules
The derivative of sin x is cos x.
sin x → cos x
And x to a power: the power rule, as before.
xn → nxn − 1
2 Each term
Use the rule on each term.
A number in front just waits, and multiplies.
f′(x) = 3x^(3 − 1) + 2 × cos x
3 Tidy up
So f′(x) = 3x² + 2 cos x.
= 3x2 + 2 cos x
4 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 4.0806, with a calculator in radians.
f′(1) = 3 + 2 cos 1 ≈ 4.0806
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 4.08.
Very close to 4.0806.
Right!
f(1.001) − f(1)0.001 ≈ 4.08
Let's look back.
First: the rules.
Then: each term.
Then: tidy up.
Last: check.
Answer
= 3x2 + 2 cos x