QCE Vault / Mathematical Methods Differentiation practice QCE Mathematical Methods · Original practice questions with worked solutions
Browse all questions Differentiation practice Integration practice Original practice exam Practise differentiation rules, tangents, rates of change and optimisation. Identify the structure of a function before choosing the product, quotient or chain rule.
Key ideas For a composite function, d d x f ( g ( x ) ) = f ′ ( g ( x ) ) g ′ ( x ) \frac{d}{dx}f(g(x))=f'(g(x))g'(x) d x d f ( g ( x )) = f ′ ( g ( x )) g ′ ( x ) . Include the derivative of the inner function. For y = u v y=uv y = uv , use y ′ = u ′ v + u v ′ y'=u'v+uv' y ′ = u ′ v + u v ′ . For y = u / v y=u/v y = u / v , use y ′ = ( u ′ v − u v ′ ) / v 2 y'=(u'v-uv')/v^2 y ′ = ( u ′ v − u v ′ ) / v 2 . A stationary point satisfies f ′ ( x ) = 0 f'(x)=0 f ′ ( x ) = 0 . Classify it using the sign of the derivative or a suitable second-derivative test, and check endpoints in optimisation. Worked example If f ( x ) = e 3 x 2 f(x)=e^{3x^2} f ( x ) = e 3 x 2 , the chain rule gives f ′ ( x ) = 6 x e 3 x 2 f'(x)=6xe^{3x^2} f ′ ( x ) = 6 x e 3 x 2 . The factor 6 x 6x 6 x comes from differentiating the exponent. A common mistake A stationary point is not automatically a maximum or minimum. Explain why your chosen point gives the required optimum on the stated domain.
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Q1 · Practice question Technology-free Simple familiar 1 mark
QUESTION 1 If f ( x ) = e 2 x f(x)=e^{2x} f ( x ) = e 2 x , then f ′ ( x ) f'(x) f ′ ( x ) is WORKED SOLUTION
Answer A 1 mark Worked solution
Using d d x e u = u ′ e u \frac{d}{dx}e^{u}=u'e^u d x d e u = u ′ e u with u = 2 x u=2x u = 2 x , f ′ ( x ) = 2 e 2 x f'(x)=2e^{2x} f ′ ( x ) = 2 e 2 x . Selects the correct option.
[1 mark] Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
View the QCAA syllabus Q26 · Practice question Technology-free Simple familiar 2 marks
QUESTION 26 (2 marks) Let f ( x ) = ln ( 2 x + 1 ) f(x)=\ln(2x+1) f ( x ) = ln ( 2 x + 1 ) . Determine f ′ ( x ) f'(x) f ′ ( x ) and hence the gradient of the tangent at x = 2 x=2 x = 2 . WORKED SOLUTION
Practice marking scheme 2 marks ANSWER f ′ ( x ) = 2 2 x + 1 f'(x)=\frac{2}{2x+1} f ′ ( x ) = 2 x + 1 2 ; gradient = 2 5 =\frac25 = 5 2 . Worked solution
Differentiate using the chain rule: f ′ ( x ) = 2 / ( 2 x + 1 ) f'(x)=2/(2x+1) f ′ ( x ) = 2/ ( 2 x + 1 ) . Substituting x = 2 x=2 x = 2 gives 2 / 5 2/5 2/5 . Determines f ′ ( x ) = 2 2 x + 1 f'(x)=\frac{2}{2x+1} f ′ ( x ) = 2 x + 1 2 . [1 mark] Determines the gradient 2 5 \frac25 5 2 . [1 mark] Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
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Q27 · Practice question Technology-free Simple familiar 2 marks
QUESTION 27 (2 marks) Differentiate y = 2 sin x + cos ( 2 x ) y=2\sin x+\cos(2x) y = 2 sin x + cos ( 2 x ) . WORKED SOLUTION
Practice marking scheme 2 marks ANSWER d y d x = 2 cos x − 2 sin ( 2 x ) \dfrac{dy}{dx}=2\cos x-2\sin(2x) d x d y = 2 cos x − 2 sin ( 2 x ) . Worked solution
d ( 2 sin x ) / d x = 2 cos x d(2\sin x)/dx=2\cos x d ( 2 sin x ) / d x = 2 cos x and d ( cos 2 x ) / d x = − 2 sin 2 x d(\cos2x)/dx=-2\sin2x d ( cos 2 x ) / d x = − 2 sin 2 x . Differentiates 2 sin x 2\sin x 2 sin x correctly. [1 mark] Differentiates cos ( 2 x ) \cos(2x) cos ( 2 x ) correctly and combines terms. [1 mark] Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
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All differentiation practice questions 98 original questions · Page 1 of 5
Q1 · Original practice · 1 mark If f ( x ) = e 2 x f(x)=e^{2x} f ( x ) = e 2 x , then f ′ ( x ) f'(x) f ′ ( x ) is Differentiation of exponential and logarithmic functions Q2 · Original practice · 1 mark Determine d d x [ tan ( 3 x ) ] \dfrac{d}{dx}[\tan(3x)] d x d [ tan ( 3 x )] . Differentiation of trigonometric functions and differentiation rules Q3 · Original practice · 1 mark The graph shown is the graph of f ′ ( x ) f'(x) f ′ ( x ) . At which value of x x x does f f f have a local maximum? Further applications of differentiation Q11 · Original practice · 1 mark If g ( x ) = ln ( 3 x 2 + 1 ) g(x)=\ln(3x^2+1) g ( x ) = ln ( 3 x 2 + 1 ) , then g ′ ( x ) g'(x) g ′ ( x ) is Differentiation of exponential and logarithmic functions Q12 · Original practice · 1 mark If h ( x ) = x 2 sin x h(x)=x^2\sin x h ( x ) = x 2 sin x , then h ′ ( x ) h'(x) h ′ ( x ) is Differentiation of trigonometric functions and differentiation rules Q13 · Original practice · 1 mark The stationary points of f ( x ) = x 3 − 3 x 2 − 9 x + 4 f(x)=x^3-3x^2-9x+4 f ( x ) = x 3 − 3 x 2 − 9 x + 4 occur when x = x= x = Further applications of differentiation Q21 · Original practice · 1 mark For f ( x ) = x ln x f(x)=x\ln x f ( x ) = x ln x , determine the equation of the tangent at x = e x=e x = e . Differentiation of exponential and logarithmic functions Q22 · Original practice · 1 mark For f ( x ) = sin x 1 + cos x f(x)=\dfrac{\sin x}{1+\cos x} f ( x ) = 1 + cos x sin x , where defined, f ′ ( x ) f'(x) f ′ ( x ) simplifies to Differentiation of trigonometric functions and differentiation rules Q23 · Original practice · 1 mark For f ( x ) = x e − x f(x)=xe^{-x} f ( x ) = x e − x on x > 0 x>0 x > 0 , the x x x -coordinate of the maximum point is Further applications of differentiation Q26 · Original practice · 2 marks Let f ( x ) = ln ( 2 x + 1 ) f(x)=\ln(2x+1) f ( x ) = ln ( 2 x + 1 ) . Determine f ′ ( x ) f'(x) f ′ ( x ) and hence the gradient of the tangent at x = 2 x=2 x = 2 . Differentiation of exponential and logarithmic functions Q27 · Original practice · 2 marks Differentiate y = 2 sin x + cos ( 2 x ) y=2\sin x+\cos(2x) y = 2 sin x + cos ( 2 x ) . Differentiation of trigonometric functions and differentiation rules Q28 · Original practice · 4 marks A rectangle is inscribed symmetrically under the parabola y = 12 − x 2 y=12-x^2 y = 12 − x 2 and above the x x x -axis, as shown. The upper right corner is ( x , 12 − x 2 ) (x,12-x^2) ( x , 12 − x 2 ) , where 0 < x < 12 0<x<\sqrt{12} 0 < x < 12 . Determine the maximum possible area of the rectangle. Further applications of differentiation Q36 · Original practice · 5 marks Consider f ( x ) = x 2 e − x f(x)=x^2e^{-x} f ( x ) = x 2 e − x . Determine all stationary points for x ≥ 0 x\ge0 x ≥ 0 and classify each as a local maximum or local minimum. Differentiation of exponential and logarithmic functions Q37 · Original practice · 5 marks For h ( x ) = sin x cos x h(x)=\sin x\cos x h ( x ) = sin x cos x on 0 ≤ x ≤ π 0\le x\le\pi 0 ≤ x ≤ π , determine the stationary points and classify them. Differentiation of trigonometric functions and differentiation rules Q38 · Original practice · 5 marks A 30 cm by 20 cm rectangular sheet has squares of side x x x cm cut from each corner. The sides are folded up to form an open box. Determine the value of x x x that maximises the volume and state the maximum volume to the nearest cubic centimetre. Further applications of differentiation Q51 · Original practice · 1 mark If f ( x ) = e 3 x − x 2 f(x)=e^{3x-x^2} f ( x ) = e 3 x − x 2 , then f ′ ( x ) f'(x) f ′ ( x ) is Differentiation of exponential and logarithmic functions Q52 · Original practice · 1 mark If y = sin ( 2 x ) + cos x y=\sin(2x)+\cos x y = sin ( 2 x ) + cos x , then d y d x \dfrac{dy}{dx} d x d y is Differentiation of trigonometric functions and differentiation rules Q53 · Original practice · 1 mark The graph of f ( x ) = x 3 − 3 x f(x)=x^3-3x f ( x ) = x 3 − 3 x is shown. The x x x -coordinates of its stationary points are Further applications of differentiation Q61 · Original practice · 1 mark If g ( x ) = ln ( e x + 1 ) g(x)=\ln(e^x+1) g ( x ) = ln ( e x + 1 ) , then g ′ ( x ) g'(x) g ′ ( x ) is Differentiation of exponential and logarithmic functions Q62 · Original practice · 1 mark If h ( x ) = cos ( 3 x ) sin x h(x)=\cos(3x)\sin x h ( x ) = cos ( 3 x ) sin x , then h ′ ( x ) h'(x) h ′ ( x ) is Differentiation of trigonometric functions and differentiation rules Q63 · Original practice · 1 mark For f ( x ) = x 4 − 4 x 2 f(x)=x^4-4x^2 f ( x ) = x 4 − 4 x 2 , the local maximum occurs at Further applications of differentiation Q71 · Original practice · 1 mark For f ( x ) = ( x + 1 ) e − x f(x)=(x+1)e^{-x} f ( x ) = ( x + 1 ) e − x , the maximum value of f f f for x ≥ 0 x\ge0 x ≥ 0 is Differentiation of exponential and logarithmic functions Q72 · Original practice · 1 mark For h ( x ) = sin x + cos ( 2 x ) h(x)=\sin x+\cos(2x) h ( x ) = sin x + cos ( 2 x ) on 0 < x < π 2 0<x<\dfrac\pi2 0 < x < 2 π , the interior stationary point occurs at Differentiation of trigonometric functions and differentiation rules Q73 · Original practice · 1 mark A rectangular enclosure beside a straight river requires fencing on only three sides, as shown. If 200 200 200 m of fencing is available, the width x x x that maximises the enclosed area is Further applications of differentiation