Differentiation

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1. 1Differentiation from first principlesWhere the derivative comes from: the gradient of a chord as the gap h shrinks to zero. Limits by cancelling and at infinity, then the four-step method for polynomials, for fractions such as 1/(2x), and for 5/(x² + 3).Done22 min
2. 2The rules, implicit differentiation and second derivativesThe quotient rule with brackets, differentiating an equation in x and y term by term, second derivatives and 'show that' questions, and tangents and normals.Done20 min

Past questions (36)

From 2008–2023: 34 WAEC, 2 NECO. Theory questions first, then objective, newest first.

Theory (27)

• WAEC 2023 · Paper 2 · Q10Calculate $\begin{vmatrix} 3 & 5 & -4 \\ 6 & -3 & -5 \\ -2 & 2 & 1 \end{vmatrix}$.
• WAEC 2022 · Paper 2 · Q10Resolve $\dfrac{x^3 - x^2 - 4}{x^2 - 1}$ into partial fractions.
• WAEC 2020 · Paper 2 · Q1If $x^2 - xy - y^2 = -31$, find $\dfrac{dy}{dx}$ at $(2, 5)$.
• WAEC 2019 · Paper 2 · Q2Differentiate from first principles, with respect to $x$, $(3x^2 + 2x - 1)$.
• WAEC 2019 · Paper 2 · Q4Differentiate, with respect to $x$, $y = 3x^2 + 4x - 1$, from first principles.
• WAEC 2018 · Paper 2 · Q11Find, from first principles, the derivative of $5x - \dfrac{6}{x^2}$ with respect to $x$.
• WAEC 2017 · Paper 2 · Q4Given that $y = 3x^2 - \dfrac{3}{x^2}$, $x \ne 0$, find:
• WAEC 2017 · Paper 2 · Q9Simplify $\dfrac{1}{1 - \cos\theta} + \dfrac{1}{1 + \cos\theta}$ and leave the answer in terms of $\sin\theta$.
• WAEC 2016 · Paper 2 · Q4Find, from first principles, the derivative of $\dfrac{5}{x^2 + 3}$ with respect to $x$.
• WAEC 2014 · Paper 2 · Q4Find the gradient of $xy^2 + x^2y = 4xy$ at the point $(1, 3)$.
• WAEC 2014 · Paper 2 · Q9Differentiate $(x - 3)(x^2 + 5)$ with respect to $x$.
• WAEC 2014 · Paper 2 · Q4Find the equation of the tangent to the curve $y = \dfrac{1}{x + 1}$ when $x = 1$.
• WAEC 2013 · Paper 2 · Q12Find the maximum and minimum points of the curve $y = 2x^3 - 3x^2 - 12x + 4$.
• WAEC 2013 · Paper 2 · Q9Differentiate $\dfrac{x^2 + 1}{(x + 1)^2}$ with respect to $x$.
• WAEC 2013 · Paper 2 · Q9Find the:
• WAEC 2012 · Paper 2 · Q10Simplify $\dfrac{4\sqrt7 + 3\sqrt2}{10 + 2\sqrt{14}}$.
• WAEC 2012 · Paper 2 · Q11If $(x + 2)$ and $(x - 1)$ are factors of $f(x) = 6x^4 + mx^3 - 13x^2 + nx + 14$, find the values of $m$ and $n$.
• WAEC 2012 · Paper 2 · Q6Find, from first principles, the derivative of $\left(3x - \dfrac{1}{2x}\right)$ with respect to $x$.
• WAEC 2011 · Paper 2 · Q12Find the equation of the tangent to the curve $\dfrac{x^2}{4} + y^2 = 1$ at the point $\left(1, \dfrac{\sqrt3}{2}\right)$.
• WAEC 2011 · Paper 2 · Q9Find, from first principles, the derivative of $f(x) = (2x + 3)^2$.
• WAEC 2011 · Paper 2 · Q10The gradient of a curve is given by $2x - 3x^2$. Find the equation of the curve if the point $(1, 2)$ lies on it.
• WAEC 2010 · Paper 2 · Q10If $x^2 + y^2 = py(1 + x^2)$, where $p$ is a constant, find $\dfrac{dy}{dx}$.
• WAEC 2010 · Paper 2 · Q12Given that $\displaystyle\int_1^m (x^2 - 2x + 1)\,dx = \dfrac13$, $m > 0$, determine the value of $m$.
• WAEC 2010 · Paper 2 · Q3Find the equation of the tangent to the curve $y = x^2 - 3x + 4$ at the point where the tangent makes an angle of $135^\circ$ with the positive …
• WAEC 2009 · Paper 2 · Q9Express $\dfrac{4x^2 - x + 3}{(x^2 + 1)(x - 1)}$ in partial fractions.
• WAEC 2008 · Paper 2 · Q3Differentiate, with respect to $x$, $x^3 + 2x$ from first principles.
• WAEC 2008 · Paper 2 · Q11Find $\dfrac{dy}{dx}$ if $x^2 + 4xy - y^2 = 7$.

Objective (9)

• NECO 2023 · Paper 1 · Q15Differentiate $\left(\dfrac{2 + y}{y}\right)^2$ with respect to $y$.
• NECO 2023 · Paper 1 · Q22If $y = x^5 + 3x^4 - 2x^2 + 2$, find $\dfrac{d^2y}{dx^2}$.
• WAEC 2023 · Paper 1 · Q13Given that $y^2 + xy = 5$, find $\dfrac{dy}{dx}$.
• WAEC 2023 · Paper 1 · Q27The distance $S$ metres moved by a body in $t$ seconds is given by $S = 5t^3 - \frac{19}{2}t^2 + 6t - 4$. Calculate the acceleration of the body …
• WAEC 2023 · Paper 1 · Q30Differentiate $f(x) = \dfrac{1}{(1 - x^2)^5}$ with respect to $x$.
• WAEC 2022 · Paper 1 · Q38Evaluate: $\displaystyle\lim_{x \to -2}\left(\frac{x^3 + 8}{x + 2}\right)$.
• WAEC 2022 · Paper 1 · Q9If $x^2 + y^2 - 2x - 6y + 5 = 0$, evaluate $\dfrac{dy}{dx}$ when $x = 3$ and $y = 2$.
• WAEC 2020 · Paper 1 · Q28A function $f$ defined by $f : x \to x^2 + px + q$ is such that $f(3) = 6$ and $f'(3) = 0$. Find the value of $q$.
• WAEC 2020 · Paper 1 · Q9Differentiate $\dfrac{x}{x + 1}$ with respect to $x$.