Simplifying before differentiating, and second derivatives

Get it into powers of x

• Expand brackets: (2x + 1)(x - 3) = 2x² - 5x - 3
• Split fractions over a single term: (x² + 6)/x = x + 6x^-1
• Roots become fractional powers: 6sqrt(x) = 6x^1/2

Never differentiate the top and bottom of a fraction separately.

Second derivative

Differentiate dy/dx again to get d^2y/dx², also written f''(x). It measures how fast the gradient is changing.

Example

y = x⁴ - 3x²: dy/dx = 4x³ - 6x and d^2y/dx² = 12x² - 6.

Worked example

y = (x³ + 4)/x². Find dy/dx and d^2y/dx².

1. Split: y = x + 4x^-2
2. dy/dx = 1 - 8x^-3
3. d^2y/dx² = 24x^-4

Answer: dy/dx = 1 - 8x^-3, d^2y/dx² = 24x^-4

Check yourself

1. y = (2x + 1)(x - 3). Find dy/dx when x = 2.
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   3. y = 2x² - 5x - 3, so dy/dx = 4x - 5 = 3 at x = 2.

2. y = (x² + 6)/x. Find dy/dx when x = 2.
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   -0.5. y = x + 6x^-1, so dy/dx = 1 - 6x^-2 = 1 - 6/4 = -0.5.

3. y = x⁴ - 3x². Find d^2y/dx² when x = 1.
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   6. dy/dx = 4x³ - 6x and d^2y/dx² = 12x² - 6 = 6 at x = 1.

4. y = 6sqrt(x). What is dy/dx?
   • 3x^-1/2
   • 3x^1/2
   • 12x^-1/2
   • 4x^3/2
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   3x^-1/2. y = 6x^1/2, so dy/dx = 6 x (1/2) x^-1/2 = 3x^-1/2.

5. f(x) = 8/x. Find f''(2).
   Show answer

   2. f(x) = 8x^-1, f'(x) = -8x^-2, f''(x) = 16x^-3 = 16/8 = 2.

6. y = x³. What is d^2y/dx²?
   • 6x
   • 3x²
   • 6
   • 6x²
   Show answer

   6x. dy/dx = 3x², then d^2y/dx² = 6x.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 1 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.