Calculating the derivative of a function of the form $\ln(ax^n+\sin(bx+c))$ using the chain rule.

Calculating the derivative of a function of the form $\ln(ax^n+e^{bx})$ using the chain rule.

Calculating the derivative of a function of the form $\ln(ax^n+bx^m)$ using the chain rule.

Calculating the derivative of a function of the form $e^{\sin(ax+b)+\cos(cx+d)}$ using the chain rule.

Calculating the derivative of a function of the form $e^{\cos(ax+b)}$ using the chain rule.

Calculating the derivative of a function of the form $e^{\sin(ax+b)}$ using the chain rule.

Calculating the derivative of a function of the form $e^{ax^n+\ln(bx)+c}$ using the chain rule.

Calculating the derivative of a function of the form $e^{ax^n+bx^m}$ using the chain rule.

Calculating the derivative of a function of the form $e^{ax^2+bx+c}$ using the chain rule.

Calculating the derivative of a function of the form $\tan(a \ln(bx))$ using the chain rule.

Calculating the derivative of a function of the form $\cos(a \ln(bx))$ using the chain rule.

Calculating the derivative of a function of the form $\sin(a \ln(bx))$ using the chain rule.

Calculating the derivative of a function of the form $\tan(e^{ax}+bx^n)$ using the chain rule.

Calculating the derivative of a function of the form $\cos(e^{ax}+bx^n)$ using the chain rule.

Calculating the derivative of a function of the form $\sin(e^{ax}+bx^n)$ using the chain rule.

Calculating the derivative of a function of the form $\tan(ax^m+bx^n)$ using the chain rule.

Calculating the derivative of a function of the form $\cos(ax^m+bx^n)$ using the chain rule.

Calculating the derivative of a function of the form $k(ax^m+b)^n$ using the chain rule.

Calculating the derivative of a function of the form $k(ax+b)^n$ using the chain rule.

Example of an explore mode question. Student is given a 2x2 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvector is recognised by the marking algorithm, also multiples of the "obvious" one(s) (given the reduced row echelon form that we use to calculate them).

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".

Calculate the functional value at a point.

Part (a): Given two cubic functions $g(x)$ and $h(x)$ of the form $ax^3+bx^2+cx+d$, give an expression for the function $f(x)$, where $f(x)=g(x)-2h(x)$. 

Part (b): Solve $f(x)=0$.

Given two cubic functions $g(x)$ and $h(x)$ of the form $ax^3+bx^2+cx+d$, solve the equation $g(x)=2h(x)$, giving all possible solutions for $x$. 

Calculating the derivative of a function of the form $(ax+b)^n$ using the chain rule.

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This question shows how to use the JSXGraph extension to show the student a plot of a function that they enter.

In the first part, the student enters an expression for $f(x)$. Once they've done that, they can reveal a plot of the function.

Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.

The answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.

This particular example has a positive gradient.

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Indefinite integration of basic functions.