1. Forgetting the inside derivative
Consider f(x) = (3x² + 1)⁴. Differentiating the outside power alone gives an incomplete answer: 4(3x² + 1)³. The inside expression also changes with x.
df/du = 4u³ and du/dx = 6x
f′(x) = 4(3x² + 1)³ × 6x
f′(x) = 24x(3x² + 1)³
A useful self-check is to name the inside function before differentiating. If the inside function is not simply x, ask whether its derivative has been included.
2. Treating every stationary point as a maximum or minimum
A stationary point has a zero first derivative. That condition alone does not tell you what happens on either side.
f′(0) = 0
For x < 0 and x > 0, the derivative is positive. The function is increasing on both sides of zero, so (0, 0) is a stationary point of inflection, not a local maximum or minimum.
Classify a stationary point by checking how the first derivative changes sign. A second derivative test can also help, but f″(a) = 0 makes that test inconclusive.
3. Losing the constant of integration
An indefinite integral describes a family of functions with the same derivative. That is why the constant matters.
If F(1) = 5, substitute the condition:
2 − 4 + C = 5, so C = 7.
The particular function is F(x) = 2x³ − 4x + 7.
Differentiate the result to check the integrand. Then use any stated condition to find C. A definite integral, evaluated between limits, gives a number and does not need a final “+ C”.
4. Confusing signed integral with total area
A definite integral counts regions below the x-axis negatively. Total geometric area counts every region positively.
The two triangular regions each have area ½. Total area is therefore ½ + ½ = 1 square unit. The signed contributions cancel in the integral; the areas do not.
Your turn: apply the chain rule
Differentiate y = (2x³ − 5)⁴, then find the gradient when x = 1.
Reveal the worked solution
= 24x²(2x³ − 5)³
At x = 1, the gradient is 24(−3)³ = −648. The factor 6x² comes from differentiating the inside expression.
For revision, keep an error log with three columns: the mistake, why it happened, and the check you will use next time. Practise the check, not only the calculation.
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