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Differentiation · Guided Practice

Differentiation

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

1
Given that y=5x3−4x+13x\displaystyle y=5x^3-4\sqrt{x}+\frac{1}{3x}, where x>0\displaystyle x > 0, find dydx\displaystyle \frac{dy}{dx}.
Express your answer without any non-integer or negative powers of x\displaystyle x.
Video solution coming soon
2
Calculate the rate of change of f(t)=t+3t, t>0\displaystyle f(t)=t+\frac{3}{t},\:t > 0, when t=2\displaystyle t=2.
Video solution coming soon
3
Find the equation of the tangent to the curve y=18x4−5\displaystyle y=\frac{1}{8} x^4-5 at the point where x=−2\displaystyle x=-2.
Video solution coming soon
4
Find the coordinates of the points on the curve y=x3−3x2\displaystyle y=x^3-3x^2 that have tangents with gradient 9\displaystyle 9.
Video solution coming soon
5
Given f(x)=5cos⁡2x\displaystyle f(x)=5 \cos 2x, evaluate f′(5π6)\displaystyle f'\left(\frac{5\pi}{6}\right).
Video solution coming soon
6
Given h(x)=2(1−4x)5, x≠14\displaystyle h(x)=\frac{2}{(1-4x)^5},\:x \neq \frac{1}{4}, find h′(18)\displaystyle h'\left(\frac{1}{8}\right).
Video solution coming soon
7
(a) Find the x\displaystyle x-coordinates of the stationary points for f(x)=x3−3x−2\displaystyle f(x)=x^3-3x-2.
(b) Hence determine the range of values of x\displaystyle x for which the function f\displaystyle f is strictly increasing.
Video solution coming soon
8
Find the dimensions of a square-based cuboid with volume 125 cm3\displaystyle 125\text{ cm}^3 and minimum surface area.
Video solution coming soon
9
Find the minimum and maximum values of f(x)=4x3+9x2−12x+1\displaystyle f(x)=4x^3+9x^2-12x+1 in the interval −1⩽x⩽2\displaystyle -1\leqslant x\leqslant 2.
Video solution coming soon
10
2017 P1 Q3
2 Marks
Given y=(4x−1)12\displaystyle y=(4x-1)^{12}, find dydx\displaystyle \frac{dy}{dx}.
11
2017 P1 Q8
3 Marks
Calculate the rate of change of
d(t)=12t\displaystyle d(t)=\frac{1}{2t}, t≠0\displaystyle t\ne0
when t=5.\displaystyle t=5.
12
2017 P2 Q7
(4, 3)7 Marks

(a)  Find the x-coordinate of the stationary point on the curve with equation
y=6x−2x3\displaystyle y=6x-2\sqrt{x^{3}}
(b)  Hence, determine the greatest and least values of y\displaystyle y in the interval 1≤x≤9\displaystyle 1\le x\le9.

13
2018 P2 Q3
3 Marks

A function, f\displaystyle f, is defined on the set of real numbers by
f(x)=x3−7x−6\displaystyle f(x)=x^{3}-7x-6
Determine whether f\displaystyle f is increasing or decreasing when x=2\displaystyle x=2.

14
2022 P1 Q12
3 Marks
Given that f(x)=4sin⁡(3x−π3)\displaystyle f(x)=4 \sin(3x-\frac{\pi}{3})
evaluate f′(π6)\displaystyle f^{\prime}(\frac{\pi}{6}).
15
2023 P1 Q1
3 Marks
Given that
y=x53−10x4\displaystyle y=x^{\frac{5}{3}}-\frac{10}{x^{4}} where x≠0\displaystyle x\ne0
find dydx\displaystyle \frac{dy}{dx}.
16
2023 P2 Q10
4 Marks

Determine the range of values of x for which the function
f(x)=2x3+9x2−24x+6\displaystyle f(x)=2x^{3}+9x^{2}-24x+6
is strictly decreasing.

17
2024 P1 Q3
2 Marks

Given that y=(5x2+3)7\displaystyle y=(5x^{2}+3)^{7}
find dydx.\displaystyle \frac{dy}{dx}.

18
2024 P1 Q12
4 Marks

The function f is given by f(x)=12x3,\displaystyle f(x)=12\sqrt[3]{x}, x>0.\displaystyle x \gt 0.
When x=a\displaystyle x=a the rate of change of f with respect to x is 1.
Determine the value of a.

19
2024 P2 Q2
5 Marks

A curve has equation
y=8x3\displaystyle y=\frac{8}{x^{3}}
where x>0.\displaystyle x \gt 0.
Find the equation of the tangent to this curve at the point where x=2\displaystyle x=2.

20
2025 P1 Q1
4 Marks
A curve has equation y=x3−2x2+5\displaystyle y=x^{3}-2x^{2}+5.
Find the equation of the tangent to this curve at the point where x=2\displaystyle x=2.
21
2026 P1 Q7
4 Marks
Determine the gradient of the tangent to the curve with equation
y=2x+4x,\displaystyle y=2x+4\sqrt{x}, x>0\displaystyle x \gt 0
at the point where x=9.\displaystyle x=9.
22
2026 P2 Q9
4 Marks
Determine the range of values of x for which the function f(x)=4x3−12x2+7\displaystyle f(x)=4x^{3}-12x^{2}+7 is strictly decreasing.
Justify your answer.
23
2026 P2 Q12
(3, 3)6 Marks

A function f is defined on the set of real numbers by
f(x)=x3+2x2−4x+1.\displaystyle f(x)=x^{3}+2x^{2}-4x+1.
(a)  Find the x-coordinates of the stationary points on the curve with equation y=f(x).\displaystyle y=f(x).
(b)  Hence determine the maximum and minimum values of f(x)\displaystyle f(x) in the interval −1≤x≤1.\displaystyle -1\le x\le 1.

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

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