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Differentiation · Guided Practice

Differentiation

25 questions with answers and video solutions. Try each one before revealing the answer.

1
Differentiate f(x)=x7tan⁡x\displaystyle f(x)=x^7\tan x.
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2
Given y=esin⁡xsec⁡x\displaystyle y=e^{\sin x}\sec x, find dydx\displaystyle \frac{dy}{dx}.
Express your answer in its simplest form.
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3
Differentiate f(x)=(ln⁡3x)(cos⁡−12x)\displaystyle f(x)=(\ln 3x)(\cos^{-1} 2x).
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4
Differentiate f(x)=2x−11−x2\displaystyle f(x)=\frac{2x-1}{1-x^2}.
Simplify the derivative fully.
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5
Differentiate f(x)=e1+x21+x2\displaystyle f(x)=\frac{e^{1+x^2}}{1+x^2}.
Express the derivative in its simplest form.
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6
Given y=ln⁡(cosec x2)\displaystyle y=\ln(\text{cosec } x^2), find dydx\displaystyle \frac{dy}{dx}.
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7
f(x)=tan⁡−1(xx3−4)\displaystyle f(x)=\tan^{-1}\left(\frac{x}{x^3-4}\right).
Find f′(2)\displaystyle f'(2).
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8
For ycot⁡x−y3=2x\displaystyle y\cot x-y^3=2x, use implicit differentiation to obtain an expression for dydx\displaystyle \frac{dy}{dx} in terms of x\displaystyle x and y\displaystyle y.
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9
Find dydx\displaystyle \frac{dy}{dx} for the function defined implicitly by xy=ey\displaystyle \frac{x}{y}=e^y.
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10
Use implicit differentiation to find dydx\displaystyle \frac{dy}{dx} and d2ydx2\displaystyle \frac{d^2 y}{dx^2} for the function defined by xy=y+1\displaystyle \frac{x}{y}=y+1.
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11
A curve is defined parametrically by x=(ln⁡t)2\displaystyle x=(\ln t)^2, y=2ln⁡t\displaystyle y=2\ln t, where t>0\displaystyle t > 0.
Find and simplify dydx\displaystyle \frac{dy}{dx} and d2ydx2\displaystyle \frac{d^2 y}{dx^2}.
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12
The position (x, y)\displaystyle (x,\,y) of a particle moving in two-dimensional space at time t\displaystyle t seconds is given in metres by the parametric equations x=2t\displaystyle x=2t, y=sin⁡t\displaystyle y=\sin t, where t⩾0\displaystyle t\geqslant 0.
Find the speed of the particle at time 2\displaystyle 2 seconds, correct to 3\displaystyle 3 significant figures.
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13
A curve is defined by y=xx2−2\displaystyle y=x^{x^2-2}.
Use logarithmic differentiation to find dydx\displaystyle \frac{dy}{dx}.
Express your answer in terms of x\displaystyle x.
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14
Let ey=(2x−1)e3x(4x+1)2\displaystyle e^y=\frac{(2x-1)e^{3x}}{(4x+1)^2}, x∈R\displaystyle x\in\mathbb{R}, x>12\displaystyle x > \frac{1}{2}.
Use logarithmic differentiation to find dydx\displaystyle \frac{dy}{dx}.
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15
A spherical balloon of radius r\displaystyle r cm is being inflated by a pump at a constant rate of 20 cm3 s−1\displaystyle 20\text{ cm}^3\text{ s}^{-1}.
Calculate the rate of change of the radius with respect to time when r=5\displaystyle r=5. [Note: a sphere has volume V=43πr3\displaystyle V=\frac{4}{3}\pi r^3.]
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16
2012 Q12
The radius of a cylindrical column of liquid is decreasing at the rate of 0.02 m s−1\displaystyle 0.02\text{ m s}^{-1} while the height is increasing at the rate of 0.01 m s−1\displaystyle 0.01\text{ m s}^{-1}.
Find the rate of change of the volume when the radius is 0.6\displaystyle 0.6 metres and the height is 2\displaystyle 2 metres. [Recall that the volume of a cylinder is given by V=πr2h\displaystyle V=\pi r^2 h.]
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17
2002 Q3
A curve is defined by the parametric equations x=t2+t−1\displaystyle x=t^2+t-1, y=2t2−t+2\displaystyle y=2t^2-t+2 for all t\displaystyle t.
Show that the point A(−1, 5)\displaystyle A(-1,\,5) lies on the curve and obtain an equation of the tangent to the curve at the point A.
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18
2019 Q10
(3, 2)5 Marks

A curve is defined implicitly by the equation x2+y2=xy+12.\displaystyle x^2 + y^2 = xy + 12.

(a)Find an expression for dydx\displaystyle \frac{dy}{dx} in terms of x\displaystyle x and y.\displaystyle y.

(b)There are two points where the tangent to the curve has equation x=k\displaystyle x = k, k∈R.\displaystyle k \in \mathbb{R}.
Find the values of k.\displaystyle k.

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19
2021 P2 Q1
2 Marks

Given f(x)=3sec⁡2x\displaystyle f(x) = 3\sec 2x find the exact value of f′(π8).\displaystyle f'\left(\frac{\pi}{8}\right).

Video solution coming soon
20
2022 P1 Q4
(3, 1, 2)6 Marks

A curve is defined by the equation y3+4y=2xy+1.\displaystyle y^3 + 4y = 2xy + 1.

(a)Use implicit differentiation to find an expression for dydx.\displaystyle \frac{dy}{dx}.

(b)Find the gradient of the tangent to the curve when y=−1.\displaystyle y = -1.

(c)Show that the curve has no stationary point.

21
2023 P2 Q1
2 Marks

The function f\displaystyle f is defined by f(x)=2sin⁡−13x.\displaystyle f(x)=2 \sin^{-1} 3x.
Find f′(x).\displaystyle f'(x).

22
2023 P2 Q10
5 Marks

A curve is defined by y=x5x2\displaystyle y=x^{5x^2}, where x>0.\displaystyle x>0.
Find dydx\displaystyle \frac{dy}{dx} in terms of x.\displaystyle x.

23
2024 P2 Q6
(3, 3)6 Marks

A curve is defined parametrically by x=t2\displaystyle x = t^2 and y=4tln⁡t\displaystyle y = 4t \ln t where t>0.\displaystyle t > 0.
Find a fully simplified expression for:

(a)dydx\displaystyle \frac{dy}{dx}

(b)d2ydx2\displaystyle \frac{d^2y}{dx^2}

24
2024 P2 Q10
4 Marks

A metal rod is heated such that its volume increases at a constant rate of 12 mm3\displaystyle 12 \text{ mm}^3 per minute.

The volume of the rod is modelled, throughout the process, by V=5πr3\displaystyle V = 5\pi r^3, where r\displaystyle r is measured in millimetres.

Find the rate at which r\displaystyle r is increasing when r=10.\displaystyle r = 10.

25
2025 P2 Q2
3 Marks

A curve is defined by the equation 2y2+4xe2y=3x.\displaystyle 2y^2 + 4xe^{2y} = 3x.
Find an expression for dydx\displaystyle \frac{dy}{dx} in terms of x\displaystyle x and y.\displaystyle y.

Practice questions courtesy of Maths.scot. Full written solutions are on his site — the links above go straight to them.

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