Express $\displaystyle \frac{a}{(x+r)(px + b)} + \frac{c}{(x+r)(qx + d)}$ as an algebraic single fraction over a common denominator. The question asks for a solution which has denominator $(x+r)(px+b)(qx+d)$.

The derivative of $\displaystyle x ^ {m}(ax^2+b)^{n}$ is of the form $\displaystyle x^{m-1}(ax^2+b)^{n-1}g(x)$. Find $g(x)$.

Differentiate $\displaystyle \ln((ax+b)^{m})$

Differentiate

\[ \sqrt{a x^m+b})\]

Differentiate $\displaystyle e^{ax^{m} +bx^2+c}$

Differentiate the following functions: $\displaystyle x ^ n \sinh(ax + b),\;\tanh(cx+d),\;\ln(\cosh(px+q))$

Differentiate $\displaystyle \cos(e^{ax}+bx^2+c)$.

Contains a video solving a similar example.

Express $\displaystyle \frac{a}{x + b} \pm  \frac{c}{x + d}$ as an algebraic single fraction over a common denominator. 

Contains a video in Show steps.

This is an old demo. For a demonstration of Numbas's current features, please see the newest demo exam.

A few questions which show off various features of Numbas.

Factorise $\displaystyle{ax ^ 2 + bx + c}$ into linear factors. 

Find $c$ and $d$ such that $x^2+ax+b = (x+c)^2+d$.

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Calculations of the lengths of two 3D vectors, the distance between their terminal points, their sum, difference, and dot and cross products.

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Product of one of 2, 3, 5, 9, or 10 by a number up to 10. With hints to learn calculation rather than memorisation.

Find $\displaystyle I=\int \frac{2 a x + b} {a x ^ 2 + b x + c}\;dx$ by substitution or otherwise.

Simple exercises introducing the fundamental set operations, and NUMBAS syntax for sets.

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