Power Rule for Differentiation
This topic covers finding the derivative (gradient function) of expressions with powers of x. For the ESAT, the main challenge is not the differentiation rule itself, but algebraically simplifying complex expressions into a simple sum of powers before you can apply it.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The core skill is simplifying first. Always expand brackets and divide through by any terms in the denominator to get a sum of `a*xn` terms.
- Rewrite roots and reciprocals as powers. For example, `√(x)` is `x^(1/2)` and `1/x3` is `x^(-3)`.
- Differentiate term by term. The derivative of a sum or difference of terms is the sum or difference of their individual derivatives.
- The derivative of any constant term (e.g., +7 or -π) is always zero.
- Advanced rules like the Chain Rule, Product Rule, and Quotient Rule are not on the ESAT specification.
Formulae
d/dx (a × xn) = a × n × x^(n-1) This is the fundamental rule for differentiating any single term where 'a' is a constant coefficient and 'n' is any rational number (positive, negative, or a fraction).
Definitions
- Derivative (dy/dx)
- A function that gives the gradient of the original function `y` at any point `x`. It represents the instantaneous rate of change.
- Rational Power
- A power `n` that can be expressed as a fraction, including integers, fractions, and negative values. For example, `x2`, `x^(1/2)`, `x^(-3)`.
Worked example
An expression is given by y = (3x + 2)2 / x^(1/2). Find dy/dx.
- 1
Step 1:
First, simplify the expression.
Avoid differentiating directly.
Expand the numerator:
(3x + 2)2 = (3x)2 + 2(3x)(2) + 22 = 9x2 + 12x + 4 - 2
Step 2:
Rewrite the expression as a sum of terms by dividing each part of the expanded numerator by the denominator, x^(1/2):
y = (9x2 / x^(1/2)) + (12x / x^(1/2)) + (4 / x^(1/2)) - 3
Step 3:
Use the laws of indices (x^a / x^b = x^(a-b)) to simplify each term:
y = 9x^(2 - 1/2) + 12x^(1 - 1/2) + 4x^(-1/2) = 9x^(3/2) + 12x^(1/2) + 4x^(-1/2) - 4
Step 4:
Now, apply the differentiation rule d/dx (a*xn) = a*n*x^(n-1) to each term individually.
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For 9x^(3/2):
(9 × 3/2) × x^(3/2 - 1) = (27/2)x^(1/2).
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For 12x^(1/2):
(12 × 1/2) × x^(1/2 - 1) = 6x^(-1/2).
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For 4x^(-1/2):
(4 × -1/2) × x^(-1/2 - 1) = -2x^(-3/2).
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Step 5:
Combine the results to get the final derivative.
Answer: dy/dx = (27/2)x^(1/2) + 6x^(-1/2) - 2x^(-3/2)
Common mistakes
- ×Trying to differentiate a fraction by differentiating the top and bottom separately. You MUST divide through first to create a sum of terms.
- ×Making arithmetic errors with fractional powers, especially when subtracting 1. For `x^(p/q)`, the new power is `(p-q)/q`.
- ×Sign errors when multiplying by a negative power. Remember `d/dx(x-3) = -3x-4`, not `3x-4`.
- ×Forgetting to multiply the coefficient by the old power. The new coefficient for `a*xn` is `a*n`, not just `a`.
No-calculator tips
- ✓To subtract 1 from a fractional power `p/q`, think of it as `p/q - q/q`. For example, `1/2 - 1` is `1/2 - 2/2 = -1/2`.
- ✓When multiplying a whole number by a fraction, multiply the whole number by the numerator only. For example, `9 × (3/2) = 27/2`.
- ✓Deal with coefficients and powers separately to avoid confusion. For `d/dx(6x^(1/3))`, first calculate the new coefficient `6 × (1/3) = 2`, then the new power `1/3 - 1 = -2/3`, and then combine them to get `2x^(-2/3)`.