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Lessons

Implicit Differentiation

Differentiating curves like $x^2+xy+y^2=12$ that can't easily be written as $y=\ldots$ : differentiate every term with respect to $x$ , remembering that $y$ terms pick up a factor $\dfrac{dy}{dx}$ .

Integration undoes differentiation: raise the power by one and divide by the new power, then add $+c$ because any constant would have vanished when differentiating. As with differentiation, rewrite roots, fractions and brackets as separate powers of $x$ first.

The Chain Rule

How to differentiate a function of a function, such as $(2x-5)^4$ or $\sqrt{x^2+5}$ , without multiplying out: differentiate the outside, keep the inside, and multiply by the derivative of the inside.

Indefinite Integration

Integration as differentiation in reverse: integrating powers of $x$ , why there is always a $+c$ , and how one point pins down the exact curve.

Gradients and the Power Rule

What the gradient of a curve means, how chords close in on the tangent, and the power rule $\dfrac{d}{dx}x^n=nx^{n-1}$ that makes differentiating fast, including roots, fractions and second derivatives.

The Product and Quotient Rules

How to differentiate two functions multiplied together, $uv$ , or divided, $\dfrac uv$ , and how to tidy the result by factorising, so it is ready for finding stationary points.

Particular Solutions with Harder Integrals

Exam differential equations are integration questions in disguise. Once the variables are separated, one side often needs partial fractions, integration by parts or a trigonometric identity. Bring all of Paper 3 integration to bear, then pin down the constant.

Syllabus topics

Topic	Board	Paper	Lesson
Implicit differentiation	Cambridge 9709	Pure Mathematics 2	Available
Modelling with differential equations	Cambridge 9709	Pure Mathematics 3	Available
Parametric differentiation	Cambridge 9709	Pure Mathematics 2	Available
Parametric differentiation	Cambridge 9709	Pure Mathematics 3	Available
Forming differential equations	Cambridge 9709	Pure Mathematics 3	Available
Implicit differentiation	Cambridge 9709	Pure Mathematics 3	Available
Variable forces and differential equations	Cambridge 9231	Further Mechanics	Available
Differentiating powers of x	Edexcel YMA01	Pure Mathematics 1	Available
Implicit differentiation	Edexcel YMA01	Pure Mathematics 4	Available
Forming differential equations	Edexcel YMA01	Pure Mathematics 4	Available
Modelling with differential equations	Edexcel YMA01	Pure Mathematics 4	Available
Parametric differentiation	Edexcel YMA01	Pure Mathematics 4	Available
Variable forces and differential equations	Edexcel YMA01	Mechanics 3	Available
Forming differential equations	Edexcel 9MA0	Pure Mathematics 1	Available
Implicit differentiation	Edexcel 9MA0	Pure Mathematics 1	Available
Implicit differentiation	Edexcel 9MA0	Pure Mathematics 2	Available
Modelling with differential equations	Edexcel 9MA0	Pure Mathematics 1	Available
Parametric differentiation	Edexcel 9MA0	Pure Mathematics 1	Available
Forming differential equations	Edexcel 9MA0	Pure Mathematics 2	Available
Modelling with differential equations	Edexcel 9MA0	Pure Mathematics 2	Available

Questions

ID	Question	Topic	Marks	Difficulty
SA-IAL-P1-DF-005	Find the gradient of y=x^3-3x^2+2 at the point where x=-1.…	IAL P1 differentiation: gradients and second derivatives Gradients at a point and rates of change	2	L1 Foundation
SA-IAL-P1-DF3-002	Find the equation of the tangent to y=x^3 at the point where x=1, in the form y=mx+c.…	IAL P1 differentiation: tangents and normals Equations of tangents	3	L1 Foundation
SA-9709-P3-DF-011	A curve is given by x=t^2, y=2t. Find the gradient of the curve at the point where t=3.…	Differentiation: parametric differentiation dy/dx for parametric curves	3	L1 Foundation
SA-9709-P2-DF-014	A curve has parametric equations x=t^2, y=2t. Find the gradient of the curve at the point where t=2.…	Differentiation: parametric differentiation dy/dx for parametric curves	3	L1 Foundation
SA-IAL-P1-DF-Q02	The tangent at a point has gradient 23. The normal has gradient…	IAL P1 differentiation: tangents and normals Equations of normals	1	L1 Foundation
SA-IAL-P1-DF2-004	Find the x-coordinate of the point on y=x^2-6x where the gradient is zero.…	IAL P1 differentiation: gradients and second derivatives Gradients at a point and rates of change	2	L1 Foundation
SA-9709-P1-DF-G01	Given y=4x^3-6x+2x, find dydx.…	Differentiation: gradients and the power rule Differentiating powers of x	3	L1 Foundation
SA-9709-P1-IN-001	Find (8x^3-6x+5)\,dx.…	Integration: reversing differentiation Integrating powers of x	3	L1 Foundation
SA-9709-P1-DF-001	Find dydx when y=3x^4-5x^2+7.…	Differentiation: gradients and the power rule Differentiating powers of x	2	L1 Foundation
SA-9709-P3-DF-Q02	For y=uv, dydx is…	Differentiation: product and quotient rules The product rule	1	L1 Foundation
SA-9709-P1-DF-Q03	At a stationary point, d^2ydx^2>0. The point is…	Differentiation: stationary points, increasing and decreasing functions Maximum or minimum: the second derivative test	1	L1 Foundation
SA-9709-P1-IN-G01	A curve has gradient dydx=6x^2-4x and passes through the point (1,5). Find the equation of the curve.…	Integration: reversing differentiation Finding a curve from its gradient and a point	4	L1 Foundation
SA-9709-P1-DF-Q02	ddx(3x+1)^5 is…	Differentiation: the chain rule The chain rule for (ax + b)ⁿ and f(x)ⁿ	1	L1 Foundation
SA-IAL-P1-DF-004	f(x)=x^3-4x. Find f''(2).…	IAL P1 differentiation: gradients and second derivatives Second derivatives	2	L1 Foundation
SA-9709-P2-DF-Q03	Differentiating y^3 with respect to x gives…	Differentiation: implicit differentiation Differentiating an equation implicitly	1	L1 Foundation