DP Mathematics SL Questionbank

• Syllabus

6.2

Sub sections and their related questions

Derivative of \({x^n}\left( {n \in \mathbb{Q}} \right)\) , \(\sin x\) , \(\cos x\) , \(\tan x\) , \({{\text{e}}^x}\) and \(\ln x\) .

• 08M.1.sl.TZ1.8a: Find \(f'(x)\) .
• 08M.2.sl.TZ1.10b(i) and (ii): (i)     Find \(f'(x)\) . (ii)    Find \(g'(x)\) .
• 12M.1.sl.TZ1.3a: Write down \(f'(x)\) .
• 12M.1.sl.TZ1.3b(i) and (ii): The tangent to the graph of f at the point \({\text{P}}(0{\text{, }}b)\) has gradient m...
• 12M.1.sl.TZ1.3c: Hence, write down the equation of this tangent.
• 10N.1.sl.TZ0.10a(i), (ii) and (iii): (i)     Show that the gradient of [PQ] is \(\frac{{{a^3}}}{{a - \frac{2}{3}}}\) . (ii)    Find...
• 10N.1.sl.TZ0.10b: Given that the area of T is \(2k + 4\) , show that k satisfies the equation...
• 10M.1.sl.TZ1.8a: Find the coordinates of A.
• 10M.1.sl.TZ1.8b(i), (ii) and (iii): Write down the coordinates of (i)     the image of B after reflection in the y-axis; (ii)   ...
• 10M.1.sl.TZ2.5: Let \(f(x) = k{x^4}\) . The point \({\text{P}}(1{\text{, }}k)\) lies on the curve of f . At P,...
• 09N.1.sl.TZ0.5a: Find \(f'(x)\) .
• 09N.1.sl.TZ0.9a: (i)     Find the coordinates of A. (ii)    Show that \(f'(x) = 0\) at A.
• 09N.2.sl.TZ0.2a: Find \(f'(x)\) .
• 09N.2.sl.TZ0.2b: Find \(g'(x)\) .
• 09M.1.sl.TZ1.8c: (i)     Find \(\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }}\) . (ii)    Hence, find the exact value of...
• 09M.1.sl.TZ2.8a: Write down (i)     \(f'(x)\) ; (ii)    \(g'(x)\) .
• 09M.2.sl.TZ2.6a: Write down the gradient of the curve at P.
• 09M.2.sl.TZ2.10d: Write down one value of x such that \(f'(x) = 0\) .
• 10M.2.sl.TZ1.9a: Show that \(A = 10\) .
• 10M.2.sl.TZ1.9b: Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of k.
• 10M.2.sl.TZ1.9c(i), (ii) and (iii): (i)     Using your value of k , find \(f'(x)\) . (ii)    Hence, explain why f is a decreasing...
• 10M.2.sl.TZ1.9d: Let \(g(x) = - {x^2} + 12x - 24\) . Find the area enclosed by the graphs of f and g .
• SPNone.1.sl.TZ0.7a: Find the first four derivatives of \(f(x)\) .
• SPNone.1.sl.TZ0.10a: Find \(f'(x)\) .
• 11N.1.sl.TZ0.9a(i), (ii) and (iii): Use the graph to write down the value of (i)     a ; (ii)    c ; (iii)   d .
• 11N.1.sl.TZ0.9b: Show that \(b = \frac{\pi }{4}\) .
• 11M.1.sl.TZ2.9a: Write down \(f(x)\) in the form \(f(x) = - 10(x - p)(x - q)\) .
• 11M.1.sl.TZ2.9b: Find another expression for \(f(x)\) in the form \(f(x) = - 10{(x - h)^2} + k\) .
• 11M.1.sl.TZ2.9c: Show that \(f(x)\) can also be written in the form \(f(x) = 240 + 20x - 10{x^2}\) .
• 11M.1.sl.TZ2.9d(i) and (ii): A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\)...
• 13M.1.sl.TZ1.3a: Find \(f'(x)\) .
• 13M.1.sl.TZ2.9a: Find \(f'(x)\) .
• 14M.1.sl.TZ1.7a: Find \(f'(x)\).
• 13N.1.sl.TZ0.10a: Show that \(f'(x) = \frac{{\ln x}}{x}\).
• 13N.2.sl.TZ0.3a: Find \(f'(x)\).
• 15M.1.sl.TZ1.9c: Find \(f'( - 2)\).
• 16M.1.sl.TZ1.9a: Find the \(x\)-coordinate of P.
• 16M.1.sl.TZ1.9b: Find \(f(x)\), expressing your answer as a single logarithm.
• 16M.1.sl.TZ1.9c: The graph of \(f\) is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\)....
• 16M.1.sl.TZ2.9b: Find \(A'(x)\).
• 16M.1.sl.TZ2.9c: Given that the outside surface area is a minimum, find the height of the container.
• 16M.1.sl.TZ2.9d: Fred paints the outside of the container. A tin of paint covers a surface area of...
• 16M.2.sl.TZ2.9a: Write down the equation of the horizontal asymptote of the graph of \(f\).
• 16M.2.sl.TZ2.9b: Find \(f'(x)\).
• 16M.2.sl.TZ2.9c: Write down the value of \(b\).
• 16M.2.sl.TZ2.9d: Given that \(g'(1) =  - e\), find the value of \(a\).
• 16M.2.sl.TZ2.9e: There is a value of \(x\), for \(1 < x < 4\), for which the graphs of \(f\) and \(g\) have...
• 16N.1.sl.TZ0.10a: (i)     Find the first four derivatives of \(f(x)\). (ii)     Find \({f^{(19)}}(x)\).
• 16N.1.sl.TZ0.10b: (i)     Find the first three derivatives of \(g(x)\). (ii)     Given that...
• 17M.1.sl.TZ2.10a.i: Write down \(f'(x)\).
• 17M.1.sl.TZ2.10a.ii: Find the gradient of \(L\).
• 17M.1.sl.TZ2.10b: Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).
• 17M.1.sl.TZ2.10c: Find the area of triangle ABC, giving your answer in terms of \(k\).
• 17M.1.sl.TZ2.10d: Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).
• 17M.2.sl.TZ1.6: Let \(f(x) = {({x^2} + 3)^7}\). Find the term in \({x^5}\) in the expansion of the derivative,...
• 17N.1.sl.TZ0.8a: Show that \(f’(1) = 1\).
• 17N.1.sl.TZ0.8b: Find the equation of \(L\) in the form \(y = ax + b\).
• 17N.1.sl.TZ0.8d: Find the area of the region enclosed by the graph of \(f\) and the line \(L\).
• 17N.1.sl.TZ0.8c: Find the \(x\)-coordinate of Q.
• 18M.2.sl.TZ1.1a: Find f '(x).
• 18M.2.sl.TZ1.1b: Find f "(x).
• 18M.2.sl.TZ1.1c: Solve f '(x) = f "(x).
• 18M.1.sl.TZ2.9a: Express h in terms of r.
• 18M.1.sl.TZ2.9b: Show that \(C = 20\pi {r^2} + \frac{{320\pi }}{r}\).
• 18M.1.sl.TZ2.9c: Given that there is a minimum value for C, find this minimum value in terms of \(\pi \).

Differentiation of a sum and a real multiple of these functions.

• 12N.1.sl.TZ0.10b: Let \(g(x) = \ln \left( {\frac{{6x}}{{x + 1}}} \right)\) , for \(x > 0\) . Show that...
• 12M.2.sl.TZ2.9a: Show that \(8a + 4b + c = 9\) .
• 12M.2.sl.TZ2.9b: The graph of f has a local minimum at \((1{\text{, }}4)\) . Find two other equations in a , b...
• 12M.2.sl.TZ2.9c: Find the value of a , of b and of c .
• 08M.1.sl.TZ1.8a: Find \(f'(x)\) .
• 12M.1.sl.TZ1.10a: Find \(s'(t)\) .
• 12M.1.sl.TZ1.10b: In this interval, there are only two values of t for which the object is not moving. One value is...
• 12M.1.sl.TZ1.10c: Show that \(s'(t) > 0\) between these two values of t .
• 12M.1.sl.TZ1.10d: Find the distance travelled between these two values of t .
• 09N.1.sl.TZ0.5a: Find \(f'(x)\) .
• 09N.1.sl.TZ0.9a: (i)     Find the coordinates of A. (ii)    Show that \(f'(x) = 0\) at A.
• 09M.1.sl.TZ1.8c: (i)     Find \(\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }}\) . (ii)    Hence, find the exact value of...
• SPNone.1.sl.TZ0.10a: Find \(f'(x)\) .
• 13M.1.sl.TZ2.9a: Find \(f'(x)\) .
• 14M.1.sl.TZ1.7a: Find \(f'(x)\).
• 18M.2.sl.TZ1.1a: Find f '(x).
• 18M.2.sl.TZ1.1b: Find f "(x).
• 18M.2.sl.TZ1.1c: Solve f '(x) = f "(x).

The chain rule for composite functions.

• 12N.1.sl.TZ0.10a: Find \(f'(x)\) .
• 12N.1.sl.TZ0.10b: Let \(g(x) = \ln \left( {\frac{{6x}}{{x + 1}}} \right)\) , for \(x > 0\) . Show that...
• 12M.2.sl.TZ2.2a: Find \(f'(x)\) .
• 12M.2.sl.TZ2.2b: On the grid below, sketch the graph of \(f'(x)\) .
• 12N.1.sl.TZ0.10c: Let \(h(x) = \frac{1}{{x(x + 1)}}\) . The area enclosed by the graph of h , the x-axis and the...
• 08N.2.sl.TZ0.9a: Show that \(f'(x) = {{\rm{e}}^{2x}}(2\cos x - \sin x)\) .
• 08M.2.sl.TZ1.10b(i) and (ii): (i)     Find \(f'(x)\) . (ii)    Find \(g'(x)\) .
• 08M.1.sl.TZ2.9b: Find \(f'(x)\) , giving your answer in the form \(a{\sin ^p}x{\cos ^q}x\) where...
• 10N.1.sl.TZ0.2a: Find \(g'(x)\) .
• 10N.1.sl.TZ0.2b: Find the gradient of the graph of g at \(x = \pi \) .
• 09N.1.sl.TZ0.9a: (i)     Find the coordinates of A. (ii)    Show that \(f'(x) = 0\) at A.
• 09N.2.sl.TZ0.2a: Find \(f'(x)\) .
• 09N.2.sl.TZ0.2b: Find \(g'(x)\) .
• 09M.1.sl.TZ2.8a: Write down (i)     \(f'(x)\) ; (ii)    \(g'(x)\) .
• 09M.2.sl.TZ2.6a: Write down the gradient of the curve at P.
• 09M.2.sl.TZ2.10d: Write down one value of x such that \(f'(x) = 0\) .
• 10M.2.sl.TZ1.9a: Show that \(A = 10\) .
• 10M.2.sl.TZ1.9b: Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of k.
• 10M.2.sl.TZ1.9c(i), (ii) and (iii): (i)     Using your value of k , find \(f'(x)\) . (ii)    Hence, explain why f is a decreasing...
• 10M.2.sl.TZ1.9d: Let \(g(x) = - {x^2} + 12x - 24\) . Find the area enclosed by the graphs of f and g .
• 11N.1.sl.TZ0.9a(i), (ii) and (iii): Use the graph to write down the value of (i)     a ; (ii)    c ; (iii)   d .
• 11N.1.sl.TZ0.9b: Show that \(b = \frac{\pi }{4}\) .
• 11N.1.sl.TZ0.9c: Find \(f'(x)\) .
• 11N.1.sl.TZ0.9d: At a point R, the gradient is \( - 2\pi \) . Find the x-coordinate of R.
• 11N.2.sl.TZ0.10a: Sketch the graph of f .
• 11N.2.sl.TZ0.10b(i) and (ii): (i)     Write down the x-coordinate of the maximum point on the graph of f . (ii)    Write down...
• 11N.2.sl.TZ0.10c: Show that \(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) .
• 11N.2.sl.TZ0.10d: Find the interval where the rate of change of f is increasing.
• 11M.2.sl.TZ1.9a: Identify the two points of inflexion.
• 11M.2.sl.TZ1.9b(i) and (ii): (i)     Find \(f'(x)\) . (ii)    Show that \(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) .
• 11M.2.sl.TZ1.9c: Find the x-coordinate of each point of inflexion.
• 11M.2.sl.TZ1.9d: Use the second derivative to show that one of these points is a point of inflexion.
• 13M.2.sl.TZ1.9d: Show that \(f'(x) = \frac{{1000{{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) .
• 13M.2.sl.TZ2.10b: Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the value of...
• 13N.1.sl.TZ0.10a: Show that \(f'(x) = \frac{{\ln x}}{x}\).
• 15N.1.sl.TZ0.10d: The following diagram shows the shaded regions \(A\), \(B\) and \(C\). The regions are...
• 16M.1.sl.TZ1.10a: Find \(f'(1)\).
• 16M.1.sl.TZ1.10b: Write down \(g'(1)\).
• 16M.1.sl.TZ1.10c: Find \(g(1)\).
• 16M.1.sl.TZ1.10d: Let \(h(x) = f(x) \times g(x)\). Find the equation of the tangent to the graph of \(h\) at the...
• 16M.2.sl.TZ2.9a: Write down the equation of the horizontal asymptote of the graph of \(f\).
• 16M.2.sl.TZ2.9b: Find \(f'(x)\).
• 16M.2.sl.TZ2.9c: Write down the value of \(b\).
• 16M.2.sl.TZ2.9d: Given that \(g'(1) =  - e\), find the value of \(a\).
• 16M.2.sl.TZ2.9e: There is a value of \(x\), for \(1 < x < 4\), for which the graphs of \(f\) and \(g\) have...
• 17M.2.sl.TZ1.6: Let \(f(x) = {({x^2} + 3)^7}\). Find the term in \({x^5}\) in the expansion of the derivative,...
• 18M.1.sl.TZ1.7: Consider f(x), g(x) and h(x), for x∈\(\mathbb{R}\) where h(x) = \(\left( {f \circ g}...
• 18M.1.sl.TZ2.10a.i: Write down \(f'\left( 2 \right)\).
• 18M.1.sl.TZ2.10a.ii: Find \(f\left( 2 \right)\).
• 18M.1.sl.TZ2.10b: Show that the graph of g has a gradient of 6 at P.
• 18M.1.sl.TZ2.10c: Let L2 be the tangent to the graph of g at P. L1 intersects L2 at the point Q. Find the...

The product and quotient rules.

• 12N.1.sl.TZ0.10a: Find \(f'(x)\) .
• 12M.1.sl.TZ2.10a: Use the quotient rule to show that \(f'(x) = \frac{{2{x^2} - 2}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\) .
• 12M.1.sl.TZ2.10b: Hence find the coordinates of B.
• 12M.1.sl.TZ2.10c: Given that the line \(y = k\) does not meet the graph of f , find the possible values of k .
• 12N.1.sl.TZ0.10c: Let \(h(x) = \frac{1}{{x(x + 1)}}\) . The area enclosed by the graph of h , the x-axis and the...
• 08N.2.sl.TZ0.9a: Show that \(f'(x) = {{\rm{e}}^{2x}}(2\cos x - \sin x)\) .
• 08M.2.sl.TZ2.9a: Show that \(f'(x) = {{\rm{e}}^x}(1 - 2x - {x^2})\) . 
• 10M.1.sl.TZ1.9a: Use the quotient rule to show that \(f'(x) = \frac{{ - 1}}{{{{\sin }^2}x}}\) .
• 10M.1.sl.TZ1.9b: Find \(f''(x)\) .
• 10M.1.sl.TZ1.9c: Find the value of p and of q.
• 10M.1.sl.TZ1.9d: Use information from the table to explain why there is a point of inflexion on the graph of f...
• 09N.2.sl.TZ0.2c: Let \(h(x) = f(x) \times g(x)\) . Find \(h'(x)\) .
• 09M.1.sl.TZ1.3: Let \(f(x) = {{\rm{e}}^x}\cos x\) . Find the gradient of the normal to the curve of f at...
• 09M.1.sl.TZ2.8b: Let \(h(x) = {{\rm{e}}^{ - 3x}}\sin \left( {x - \frac{\pi }{3}} \right)\) . Find the exact value...
• 10M.2.sl.TZ1.3a: Find \(f'(x)\) .
• 10M.2.sl.TZ1.3b: On the grid below, sketch the graph of \(y = f'(x)\) .
• 10M.2.sl.TZ2.10a(i) and (ii): Let P and Q be points on the curve of f where the tangent to the graph of f is parallel to the...
• 10M.2.sl.TZ2.10b: Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . Show that...
• 10M.2.sl.TZ2.10c: Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . Sketch the graph of \(g'\) .
• 10M.2.sl.TZ2.10d: Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . Consider \(g'(x) = w\) ....
• 11N.2.sl.TZ0.10a: Sketch the graph of f .
• 11N.2.sl.TZ0.10b(i) and (ii): (i)     Write down the x-coordinate of the maximum point on the graph of f . (ii)    Write down...
• 11N.2.sl.TZ0.10c: Show that \(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) .
• 11N.2.sl.TZ0.10d: Find the interval where the rate of change of f is increasing.
• 11M.1.sl.TZ1.5a: Use the quotient rule to show that \(g'(x) = \frac{{1 - 2\ln x}}{{{x^3}}}\) .
• 11M.1.sl.TZ1.5b: The graph of g has a maximum point at A. Find the x-coordinate of A.
• 11M.1.sl.TZ2.4: Let \(h(x) = \frac{{6x}}{{\cos x}}\) . Find \(h'(0)\) .
• 13M.1.sl.TZ1.3a: Find \(f'(x)\) .
• 13M.1.sl.TZ2.10d: find the equation of the normal to the graph of \(h\) at P.
• 14M.2.sl.TZ1.7: Let \(f(x) = \frac{{g(x)}}{{h(x)}}\), where \(g(2) = 18,{\text{ }}h(2) = 6,{\text{ }}g'(2) = 5\),...
• 14M.1.sl.TZ2.10a: Use the quotient rule to show that \(f'(x) = \frac{{10 - 2{x^2}}}{{{{({x^2} + 5)}^2}}}\).
• 16M.1.sl.TZ1.10a: Find \(f'(1)\).
• 16M.1.sl.TZ1.10b: Write down \(g'(1)\).
• 16M.1.sl.TZ1.10c: Find \(g(1)\).
• 16M.1.sl.TZ1.10d: Let \(h(x) = f(x) \times g(x)\). Find the equation of the tangent to the graph of \(h\) at the...
• 16N.1.sl.TZ0.10c: (i)     Find \(h'(x)\). (ii)     Hence, show that \(h'(\pi ) = \frac{{ - 21!}}{2}{\pi ^2}\).
• 17M.1.sl.TZ2.6a: Find \(h(1)\).
• 17M.1.sl.TZ2.6b: Find \(h'(8)\).

The second derivative.

• 10M.1.sl.TZ1.9a: Use the quotient rule to show that \(f'(x) = \frac{{ - 1}}{{{{\sin }^2}x}}\) .
• 10M.1.sl.TZ1.9b: Find \(f''(x)\) .
• 10M.1.sl.TZ1.9c: Find the value of p and of q.
• 10M.1.sl.TZ1.9d: Use information from the table to explain why there is a point of inflexion on the graph of f...
• 09M.1.sl.TZ2.6a: Find the second derivative.
• 09M.1.sl.TZ2.6b: Find \(f'(3)\) and \(f''(3)\) .
• 11M.2.sl.TZ1.9a: Identify the two points of inflexion.
• 11M.2.sl.TZ1.9b(i) and (ii): (i)     Find \(f'(x)\) . (ii)    Show that \(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) .
• 11M.2.sl.TZ1.9c: Find the x-coordinate of each point of inflexion.
• 11M.2.sl.TZ1.9d: Use the second derivative to show that one of these points is a point of inflexion.
• 13M.1.sl.TZ1.10c: (i)     Find \(f''(1)\) . (ii)     Hence, show that there is no point of inflexion on the graph...
• 15M.1.sl.TZ1.9a: Find \(f''(x)\).
• 16N.1.sl.TZ0.10a: (i)     Find the first four derivatives of \(f(x)\). (ii)     Find \({f^{(19)}}(x)\).
• 18M.1.sl.TZ1.8a: Find f (x).
• 18M.1.sl.TZ1.8b: The graph of f has a point of inflexion at x = p. Find p.
• 18M.1.sl.TZ1.8c: Find the values of x for which the graph of f is concave-down.

Extension to higher derivatives.

• SPNone.1.sl.TZ0.7b: Write an expression for \({f^{(n)}}(x)\) in terms of x and n .
• 16N.1.sl.TZ0.10a: (i)     Find the first four derivatives of \(f(x)\). (ii)     Find \({f^{(19)}}(x)\).
• 18M.1.sl.TZ1.8a: Find f (x).
• 18M.1.sl.TZ1.8b: The graph of f has a point of inflexion at x = p. Find p.
• 18M.1.sl.TZ1.8c: Find the values of x for which the graph of f is concave-down.