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.

Solving a differential equation of the form $\frac{dy}{dx}=a \cos(x) e^{-y}$ using separation of variables.

Solving a differential equation of the form $\frac{dy}{dx}=ax^n e^{-y}$ using separation of variables.

Solving a differential equation of the form $\frac{dy}{dx}=\frac{a \cos(x)}{y}$ using separation of variables.

Solving a differential equation of the form $\frac{dy}{dx}=x(y-a)$ using separation of variables.

Rewriting fractions involving surds by rationalising the denominator.

Rewriting fractions involving surds by rationalising the denominator.

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.

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Unit rates and converting rates.