Power Rule for Differentiation
The power rule finds the derivative (gradient function) of expressions with powers of x; for the ESAT, the main challenge is not the 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.
- 5
For 9x^(3/2):
(9 × 3/2) × x^(3/2 - 1) = (27/2)x^(1/2).
- 6
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)`.