Chain Rule & Integration by Substitution
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What is the rate of change of a composite function?

Key questions

  1. 1

    How is the rate of change of \(f(g(x))\) related to \(f'(x)\) and \(g'(x)\)?

  2. 2

    When and how can the inverse of a function be differentiated?

  3. 3

    What is implicit differentiation?

  4. 4

    How can gradients of parametric curves be found?

  5. 5

    What is integration by substitution, and how is it related to the chain rule?

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Introducing...

Resource type Title
Building blocks Chain mapping
Building blocks Parametric preliminaries
Building blocks Reflecting on change
Building blocks Slippery areas
Building blocks Slippery slopes... another derivative
Many ways problem Implicit circles

Developing...

Resource type Title
Rich example Circular slice
Rich example Which substitution?
Building blocks Powerful derivatives
Fluency exercise I can see $u$!
Fluency exercise Integral sorting
Fluency exercise Parametric paths
Fluency exercise Parametric points
Package of problems Odd one out
Many ways problem Can you find... chain rule edition
Scaffolded task Differentiating exponentials
Bigger picture The law of reflection and saving time

Review questions

Title Ref
Can we carry out these three tricky integrals? R8778
Can we find $\int 1/(\sqrt{x+2} - \sqrt{x+1}) dx$? R9701
Can we find a tangent to $y=1/(1+x^2)$? R9085
Can we find the angle between these two tangents? R5678
Can we find the equation of the normal to the curve when $t = 2$? R8620
Can we find the minimum value of $x^2+2/x$ for $x$ positive? R5640
Can we find this area by integrating with respect to $y$? R9815
Can we find where the normal is parallel to $y=x$? R8939
Can we show $x>\ln(1+x^2)$ for all $x>0$? R7431
Can we show that these normals intersect on this circle? R9413
Can we show that these normals meet on the $x$-axis? R5696
Can we show these tangents to this hyperbola form a square? R7833
Can we show this parabola and this ellipse meet at right angles? R7042
Can we sketch $y = x^2(2-x)$, and from this, $y^2 = x^2(2-x)$? R7839
Can we sketch $y=3 \cos x - \cos 3x$? R5251
Can we use integration by substitution on these integrals? R8890
How could we integrate $\tan^3 \theta$? R8694
How could we integrate $\tan^4x$? R9297
How many normals to a parabola pass through $P$? R5053
How many real roots does $(x-1)(x+2)(x+3)=3x$ have? R7792
If we know the object's speed, can we find the speed of the image? R8625
What can we say if a normal to an ellipse passes through this point? R5237
What can we say if a point $P$ on an ellipse is directly above the focus? R6710
What can we say if this hyperbola and parabola meet at right angles? R5376
What happens when we differentiate $\sin^4x + 2 \cos^2x - \cos^4x$? R5990
What is implied if these two parabolas touch exactly once? R5739
When do these normals to a parabola meet on the parabola? R9809
When does this cone have a maximum volume? R7115
When is the volume of this pyramid a maximum? R9905
Where does this chord of an ellipse cut the $x$-axis? R6843
Where does this tangent to this ellipse meet $x=a$? R9325
Where is the centre of mass of half this cylinder? R8836
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