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## When and how can we differentiate the product or quotient of two functions?

### Key questions

1. 1

How does the area of a rectangle change when we vary the lengths of the sides?

2. 2

How can we differentiate the product of two functions?

3. 3

Do we need a rule for differentiating a quotient of functions?

4. 4

What do we mean by integration by parts, and when is it useful?

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#### Introducing...

Resource type Title
Building blocks Producing a rule
Scaffolded task How does your rectangle grow?

#### Developing...

Resource type Title
Fluency exercise Integral chasing II
Many ways problem Constant consistency

#### Review questions

Title Ref
Can we differentiate $e^{2x}( 2 \cos (3x) + 3 \sin (3x))$? R6530
Can we find all three integrals? R5126
Can we find the area between $y=e^{-x}$ and $y=e^{-x}\sin x$? R9312
Can we find the repeated root for $4x^4+x^2+3x+1=0$? R9082
Can we find this body's displacement from $O$ when its velocity is $0$? R8774
Can we show the function $(2x+1)/(x^2-1)$ can take all real values? R9329
Can we sketch $y=ax/(x^2+x+1)$ when $a$ is positive? R7129
Can we sketch $y=x^2/(1+x^4)$? R5501
Can we sketch the curve $y=(x-a)e^x/(x-b)$? R9898
Can we sketch the curve $y=1/((x-a)^2-1)$? R6855
Can we sketch the curve $y=e^{-x}/(1+x^2)$? R9484
Can we sketch the graph of $y=x(x+1)(x-2)^4$? R8387
Given that $y = (x+6)^7(x-9)^8$, can we find $dy/dx$? R9795
How could we integrate $e^{-x}\sin^n x$? R8134
If $dy/dx$ is inversely proportional to $x^2$, can we find $y$? R7410
If $y=(x^2+1)/(x^2-a^2)$, then $y$ cannot take which values? R8765
If the gradient is $x(1-x^2)e^{-x^2}$ can we find the stationary points? R6620
If this cubic has two equal roots, what can $p$ be? R5196
If we can sketch $y = f(x)$, can we sketch $y^2 = f(x)$? R6863
What is the average density of this spherical planet? R8152
When will the arclength $QR$ take a maximum value? R8936
Where are the stationary points on $\cot x-8\cos x$? R6549
Which is the greater of $e^{\pi}$ and $\pi^e$? R8671
Which values can $x(x-3)/(x-4)$ take? R9391