IB Maths AA HL Sl 5 6 Differentiation Rules Questions

Practise differentiating powers, trigonometric, exponential and logarithmic functions with sum, chain, product and quotient rules, then interpreting the resulting rate.

Exam points

Apply the chain, product and quotient rules to differentiate composite, product and quotient functions, including exponential, logarithmic and trigonometric forms.
Use implicit differentiation, inverse-function relationships and related rates to obtain and interpret derivatives.
Evaluate first and higher derivatives at specified inputs and interpret them as gradients, rates of change, velocity or acceleration.
Use derivatives to solve stationary-point, inflexion, maximum-rate and parameter problems, and to verify differential relationships.

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Study SL 5.6—Differentiation rules conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 5]

The following question explores features of a family of curves. The family is then linked to a homogeneous differential equation.
Consider the curve given by $y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ .

(a)

Given $f(x)=\frac{x\left(x^{2}-A\right)}{x^{2}+A}$ , prove that $f^{\prime}(\sqrt{A})$ is independent of A.

[ 4 ]

attempt to use quotient rule or product rule
$f^{\prime}(x)=\frac{\left(x^{2}+A\right)\left(3 x^{2}-A\right)-2 x\left(x^{3}-A x\right)}{\left(x^{2}+A\right)^{2}} \quad\left(f^{\prime}(x)=\frac{2 x^{2}+\left(x^{2}-A\right)}{x^{2}+A}-\frac{2 x^{2}\left(x^{2}-A\right)}{\left(x^{2}+A\right)^{2}}\right)$
(or equivalent)

Note: Award A1 for numerator, A1 for denominator.
Note: If product rule used, award A1 for each correct term.
Note: Award (M1) for an attempt to use quotient rule or product rule with A=16 or any other constant. Award no further marks.

f^{\prime}(\sqrt{A})\left(=\frac{(2 A)(2 A)-0}{(2 A)^{2}}\right)=1

hence independent of A

(b)

Write down the coordinates of a point on the curve where the oblique asymptote is parallel to the tangent to the curve at that point.

Now consider the differential equation $x^{2} \frac{\mathrm{~d} y}{\mathrm{~d} x}=x(x+y)-y^{2}$ , where $x \neq 0, y \neq \pm x$ .
Using the substitution y=v x, the differential equation can be written as $x \frac{\mathrm{~d} v}{\mathrm{~d} x}=1-v^{2}$ .

[ 1 ]

$(\sqrt{A}, 0) \mathrm{OR}(-\sqrt{A}, 0)$

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IB Maths AA HL Sl 5 6 Differentiation Rules Questions question 1

Question (a)

Question (b)