Rewrite roots and fractions as powers before differentiating. √x = x^(1/2), so its derivative is ½x^(−1/2). 6/√x = 6x^(−1/2), so its derivative is −3x^(−3/2). Expand any brackets first: √x(x − 5) = x^(3/2) − 5x^(1/2).
The second derivative d²y/dx² is the derivative of the derivative. It tells you how the gradient is changing. For y = 4x³ − 7x², dy/dx = 12x² − 14x and d²y/dx² = 24x − 14.
Worked example
Find dy/dx when y = 5x − 9 + 6/√x.
- Write 6/√x as 6x^(−1/2).
- dy/dx = 5 + 6 × (−½)x^(−3/2).
Answer:dy/dx = 5 − 3x^(−3/2)
Key words
- second derivative
- the derivative of dy/dx
- negative power
- a reciprocal, like x⁻² = 1/x²
- fractional power
- a root, like x^(1/2) = √x
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