Step by step
The question
Question: find dy/dx for y = 2 x⁴ + 3 x².
y = 2x4 + 3x2
1 The rule
From the definition we found: x² gives 2x, and x³ gives 3x².
x2 → 2x
x3 → 3x2
See the pattern?
The power comes down in front, and the power goes down by 1.
This is the power rule.
ddxxn = nxn − 1
dy/dx is another way to write the derivative of y.
Say it "d y by d x".
dy/dx: the derivative of y
2 Each term
Use the rule on each term.
A number in front just waits, and multiplies.
dydx = 2 × 4x^(4 − 1) + 3 × 2x^(2 − 1)
Careful!
The power must go down by 1 too.
Careful: not 2x4 → 8x4 ✗
Work out each one.
So dy/dx = 8 x³ + 6x.
= 8x3 + 6x
3 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 14.
dy/dx at x = 1: 8 + 6 = 14
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 14.02.
Very close to 14.
Right!
y(1.001) − y(1)0.001 ≈ 14.02
Let's look back.
First: the rule.
Then: each term.
Last: check.
Answer
= 8x3 + 6x