To differentiate any term axn, multiply by the power and then reduce the power by one: d/dx of axn = anxn−1. This holds when n is negative or a fraction as well. The first job is to rewrite roots and fractions as powers of x.
This is the base skill of differentiation techniques. Every later rule finishes by applying it.
How do you turn roots and fractions into powers?
Use the index laws first, then differentiate. The common conversions are:
| Written as | Power form |
|---|---|
| √x | x1/2 |
| 1/x | x−1 |
| 1/x² | x−2 |
| 1/√x | x−1/2 |
| ∛x | x1/3 |
| (x² + 3)/x | x + 3x−1 |
A fraction with a single term underneath can always be split. Divide each term on top by the bottom, then simplify each one using the index laws.
Worked example
Differentiate y = 2x³ − 6√x + 4/x², and find the gradient at x = 4.
Step 1, rewrite: y = 2x³ − 6x1/2 + 4x−2.
Step 2, differentiate term by term:
- 2x³ gives 6x².
- −6x1/2 gives −6 × ½ × x−1/2 = −3x−1/2.
- 4x−2 gives 4 × (−2) × x−3 = −8x−3.
So dy/dx = 6x² − 3x−1/2 − 8x−3 = 6x² − 3/√x − 8/x³.
Step 3, substitute x = 4: 6 × 16 = 96, then 3/√4 = 3/2 = 1.5, then 8/4³ = 8/64 = 0.125.
Gradient = 96 − 1.5 − 0.125 = 94.375.
The mistake to watch for
A common slip is to reduce the power from the wrong starting value, especially when it is already negative.
Mistaken working: y = 4/x² → dy/dx = 4 × (−2) × x−1 = −8/x
The student treated −2 − 1 as −1.
The power starts at −2, and reducing it by one gives −3, not −1. So the correct derivative is −8x−3 = −8/x³. Writing “new power = old power − 1” on your line, with the arithmetic shown, prevents this.
The same care applies to fractions: x1/2 becomes x−1/2, because ½ − 1 = −½.
Check yourself
1. Differentiate y = x3/2.
Show answer
Bring down 3/2 and reduce the power: 3/2 − 1 = 1/2.
dy/dx = (3/2)x1/2
2. Differentiate y = 5/√x and find the gradient at x = 4.
Show answer
Rewrite: y = 5x−1/2. Then dy/dx = 5 × (−½) × x−3/2 = −(5/2)x−3/2.
At x = 4, x3/2 = 8, so the gradient is −5/(2 × 8) = −5/16.
3. Differentiate y = (x² + 3)/x and find the gradient at x = 3.
Show answer
Split the fraction: y = x + 3x−1. Then dy/dx = 1 − 3x−2 = 1 − 3/x².
At x = 3: 1 − 3/9 = 1 − 1/3 = 2/3.
Where this leads next
With powers secure, move on to differentiating a product, where each factor is differentiated with this same rule. The calculus shape and rate explorer shows how the gradient formula relates to the curve, and the non-calculator working trainer builds the fluency for fractions and roots.
Students who know the rule but drop signs or indices under time pressure are the ones our teachers spend the most time with in online one-to-one Additional Mathematics tuition.