Differentiating powers of x

In 5.1 the gradient was a limit you estimated from a table. Nobody wants to build a table every time, so there is a rule. Press play and watch it happen to one term.

3x⁴ start One term. The power is 4 and the coefficient is 3.

3x⁴ → 12x⁴ Move 1. Multiply the front by the power: 4 × 3 = 12.

12x⁴ → 12x³ Move 2. Knock one off the power: 4 − 1 = 3.

If f(x) = axⁿ then f′(x) = anxⁿ⁻¹

Two moves, always in that order.

Every term, one at a time

A polynomial is just several of those, and you do each one separately. Press play and watch each term get the same treatment.

f(x) = 3x⁴ − 5x² + 7x − 2

3x⁴→ 12x³ 4 × 3 = 12, and the power drops to 3.

−5x²→ −10x 2 × −5 = −10, and x¹ is just x.

7x→ 7 7x is 7x¹, so 1 × 7 = 7 and x⁰ = 1. A straight line has a constant gradient.

−2→ 0 A constant has no slope. Shifting a graph up or down never changes its gradient.

f′(x) = 12x³ − 10x + 7

Signs travel with their term.

Does the rule actually give the gradient?

Fair question, since 5.1 said a gradient was a limit. So let us check. On the left is the rule. On the right is a real chord across a tiny gap, worked out the 5.1 way. Move x and watch them agree.

g(x) = 2x³ + x² − 4x + 9 so g′(x) = 6x² + 2x − 4

x = 1.00

4.000from the rule

4.000from a tiny chord

The chord uses a gap of 0.0001 either side. They agree to three decimal places everywhere, which is the point.

Negative powers are in. Roots are not.

On Applications and Interpretation the power n is a whole number. It can be negative, so ¹⁄_x² is fair game. It is never a fraction, so you are not asked to differentiate √x on this course.

If you are using a textbook or a video made for Analysis and Approaches you will meet √x = x^1/2 differentiated as if it were routine. For AI that is extra work you cannot be examined on.

Worked example: a negative power

Differentiate h(x) = 4x².

h(x)= 4x⁻² rewrite it as a power first, always h′(x)= −2 × 4x⁻²⁻¹ = −8x⁻³ the rule, unchanged = −8x³ back to a fraction if the question used one

Check it at x = 2: −8 ÷ 8 = −1. Negative, which fits, because 4⁄x² is falling at x = 2.

Knocking one off a negative power makes it more negative. −2 − 1 = −3, not −1. This is the single most common slip on this sub-topic, and it is a subtraction error rather than a calculus one.

Your turn

1. For f(x) = 3x⁴ − 5x² + 7x − 2, find f′(1).

2. Differentiate 4x².

−8x⁻³ −8x⁻¹ 8x⁻³ −2x⁻³

3. Which of these are you not expected to differentiate on Applications and Interpretation?

x⁻³ √x 5x⁷ −9

Where the marks go

Rewrite fractions as powers before you differentiate. Trying to differentiate 4⁄x² while it still looks like a fraction is where most wrong answers begin.

Do not lose the constant term by forgetting it exists. It differentiates to zero, which is a thing you did, not a thing you skipped.

If the question says "hence" it wants you to use the derivative you just found. A fresh calculator answer that ignores your own working can cost the method marks even when the number is right.

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