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ScotlandMaths

Methods in Algebra and Calculus

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you differentiate products, quotients, composite, inverse, implicit and parametric functions?

Differentiate using the chain, product and quotient rules; differentiate exponential, logarithmic, inverse trigonometric, implicit and parametrically defined functions; and use logarithmic differentiation and higher derivatives.
A focused answer to the SQA Advanced Higher Mathematics differentiation techniques content, covering the chain, product and quotient rules, differentiation of exponential, logarithmic and inverse trigonometric functions, implicit and parametric differentiation, logarithmic differentiation, and the second derivative.
12 min answer →

How do you solve a first-order differential equation that is separable or linear?

Solve first-order differential equations by separating the variables and by the integrating-factor method for linear equations, find particular solutions from initial conditions, and apply differential equations to growth and decay models.
A focused answer to the SQA Advanced Higher Mathematics first-order differential equations content, covering separable equations, the integrating-factor method for linear first-order equations, finding particular solutions from initial conditions, and applications to growth and decay.
11 min answer →

Which technique do you reach for when an integral is not a standard result?

Integrate using standard results, integration by substitution, integration by parts and integration using partial fractions, and apply integration to find areas and volumes of revolution.
A focused answer to the SQA Advanced Higher Mathematics integration techniques content, covering standard results, integration by substitution, integration by parts, integration using partial fractions, and applications to areas and volumes of revolution.
12 min answer →

How do you split a rational function into simpler fractions so it can be integrated or expanded?

Express a proper rational function as a sum of partial fractions where the denominator factorises into distinct linear factors, repeated linear factors, or an irreducible quadratic factor, and reduce an improper rational function first by algebraic division.
A focused answer to the SQA Advanced Higher Mathematics partial fractions content, covering proper and improper rational functions, denominators with distinct linear factors, repeated linear factors and an irreducible quadratic factor, and the cover-up and equating-coefficients methods used to find the constants.
11 min answer →

How do you solve a linear second-order differential equation with constant coefficients?

Solve homogeneous and non-homogeneous second-order linear differential equations with constant coefficients using the auxiliary equation, the complementary function and a particular integral, covering distinct real, equal and complex roots.
A focused answer to the SQA Advanced Higher Mathematics second-order differential equations content, covering the auxiliary equation, the three cases of distinct real, equal and complex roots, the complementary function, finding a particular integral for non-homogeneous equations, and the general solution.
12 min answer →