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SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules - IB Questionbank | RevisionDojo

Practice SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider the function $g(x) = e^x \cos(x)$ ,.

Find the derivative of the function $g(x) = e^x \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Find the equation of the normal line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Question 2

SLPaper 1

The function j is defined for all x ∈ ℝ. The line with equation y = 6x - 1 is the tangent to the graph of j at x = 4.

Write down the value of j'(4).

[1]

Find j(4).

[1]

The function k is defined for all x ∈ ℝ where k(x) = x² - 3x and m(x) = j(k(x)). Find m(4).

[2]

Hence find the equation of the tangent to the graph of m at x = 4.

[3]

Question 3

SLPaper 1

The equation of a curve is $y = \frac{1}{2}x^4 - \frac{3}{2}x^2 + 7$ .

The gradient of the tangent to the curve at a location Q is $-10$ .

Find ${\frac{{dy}}{{dx}}}$ .

[2]

Find the coordinates of Q.

[4]

Question 4

SL & HLPaper 1

Consider the function $f(x) = \sin(x) + \cos(x)$ , where $x$ is in radians.

Find the derivative of the function $f(x) = \sin(x) + \cos(x)$ .

[2]

Determine the critical points of the function $f(x) = \sin(x) + \cos(x)$ in the interval $[0, 2\pi]$ .

[3]

Classify the critical points found in part (b) as local maxima, minima, or inflection points.

[3]

Question 5

SLPaper 1

Let $g'(x) = \frac{{8x}}{{\sqrt{{2x^2 + 1}}}}$ . Given that $g(0) = 5$ , find $g(x)$ .

[5]

Question 6

HLPaper 1

Consider the homogeneous differential equation $\frac{dy}{dx} = \frac{y^2-2x^2}{xy}$ , where $x, y \neq 0$ . It is given that $y = 2$ when $x = 1$ .

By using the substitution $y = vx$ , solve the differential equation. Give your answer in the form $y^2 = f(x)$ .

[8]

The points of zero gradient on the curve $y^2 = f(x)$ lie on two straight lines of the form $y = mx$ where $m \in \mathbb{R}$ . Find the values of $m$ .

[2]

Question 7

SL & HLPaper 1

Consider the function $g(x)=\frac{\sin(2x)}{2}(\frac{1}{\tan(x)}+2)-\sin^2(x)$

Show that $g(x)=\sin(2x)+\cos(2x)$

[5]

Show that $g''(x)=-4g(x)$

[3]

Hence, using your answer from part (b), find $\int\!g(x)\,dx$ in terms of $g(x)$

[4]

Question 8

SL & HLPaper 1

Given the function $g(x) = \tan(x)$ , where $x$ is in radians and $x \neq \frac{\pi}{2} + n\pi$ , $n \in \mathbb{Z}$ .

Find the derivative of the function $g(x) = \tan(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = \tan(x)$ at the point where $x = \frac{\pi}{4}$ .

[4]

Question 9

SLPaper 1

A sculptor sells $x$ sculptures per month. Her monthly profit in Canadian dollars (CAD) can be modelled by $P\left(x\right) = -\frac{1}{5}x^3 + 7x^2 - 120,\,\,x \geqslant 0.$

Differentiate $P(x)$ .

[2]

Hence, find the number of sculptures that will maximize the profit.

[3]

Find the value of $P$ if no sculptures are sold.

[1]

Question 10

HLPaper 1

In this question, all lengths are in metres and time is in seconds. Consider two particles, $A_1$ and $A_2$ , which start to move at the same time. Particle $A_1$ moves in a straight line such that its displacement from a fixed-point is given by $s(t) = 10 - \frac{7}{4}t^2$ , for $t \geq 0$ .

Find an expression for the velocity of ${A}_1$ at time ${t}$ .

[2]

Particle $A_2$ also moves in a straight line. The position of $A_2$ is given by $r=-\frac{1}{6}+t\left(\begin{array}{c}4\\-3\end{array}\right).$ The speed of $A_1$ is greater than the speed of $A_2$ when $t>q$ . Find the value of $q$ .

[5]

SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules Questionbank