Differentiation

Quick review

1. Find derivatives of the following functions:
   

   (a)   $f(x)=x^4-2x^3+x^2-5x+7$

   (b)   $f(x)=\frac{1}{x^2}+\sqrt{x^3}$

   (c)   $f(x)=2e^x+\ln x+\cos x$

   Solutions:    (a)  $f^{\prime }(x)=4x^3-6x^2+2x-5$;     (b)  $f^{\prime }(x)=-\frac{2}{x^3}+\frac{3}{2}\sqrt{x}$;     (c)  $f^{\prime }(x)=2e^x+\frac{1}{x}-\sin x$
2. Differentiate the following functions:
   

   (a)   $f(x)=x^3\ln x$

   (b)   $f(x)=e^x\sin x$

   (c)   $f(x)=\frac{2x+1}{x-2}$

   (d)   $f(x)=\frac{\sin x+2}{\cos x}$

   Solutions:    (a)  $f^{\prime }(x)=3x^2\ln x+x^2$;     (b)  $f^{\prime }(x)=e^x\sin x+e^x\cos x$;     (c)  $f^{\prime }(x)=\frac{-5}{x^2-4x+4}$;     (d)  $f^{\prime }(x)=\frac{1+2\sin x}{{\cos }^{2}x}$
3. Differentiate the following functions:
   

   (a)   $y=(4x+1{)}^3$

   (b)   $y=\sqrt{x^2+9}$

   (c)   $y=\sin \frac{x+\pi }{5}$

   Solutions:    (a)  $y^{\prime }=3(4x+1{)}^2\cdot 4=12(4x+1{)}^2$;     (b)  $y^{\prime }=\frac{x}{\sqrt{x^2+9}}$;     (c)  $y^{\prime }=\frac{1}{5}\cos \frac{x+\pi }{5}$
4. The function has the equation:  $f(x)=\frac{x^2+x+1}{3}$.
   

   (a)   Differentiate the function $f$.

   (b)   Write down the equation of the tangent at $x=4$.

   (c)   Show that this tangent is parallel to the straight line $y=3x+1$.

   Solutions:    (a)  $f^{\prime }(x)=\frac{2x+1}{3}$;     (b)  tangent: $y=3x-5$;     (c)  they have the same gradient: $m_1=m_2=3$
5. The function has the equation:  $f(x)=\sqrt{2x+7}$.
   

   (a)   Differentiate the function $f$.

   (b)   Write down the equation of the normal at $x=1$.

   (c)   This normal and both coordinate axes form a triangle:

   (i)   Write down the coordinates of the vertices of this triangle.

   (ii)   Calculate the area of this triangle.

   Solutions:    (a)  $f^{\prime }(x)=\frac{1}{\sqrt{2x+7}}$;     (b)  normal: $y=-3x+6$;     (c)  (i)  vertices: $A(0,0),B(2,0),C(0,6)$; (ii)  area: $A=6$

Stationary points

6. The function has the equation $f(x)=2x^3-15x^2+36x-25$. Find the points on the graph where the tangent is horizontal. Write down the coordinates of these points.
   Solutions:    $P_1(2,3),P_2(3,2)$
7. The function has the equation $f(x)=x^3-3x+1$. Find the stationary points of this function. Write down the coordinates of these points.
   Solutions:    $P_1(-1,3),P_2(1,-1)$
8. The function has the equation $f(x)=x^4-4x^2$.
   

   (a)   Find the zeros of this function.

   (b)   Find the stationary points of this function.

   (c)   Draw the graph.

   Solutions:    (a)  $x_1=-2,x_{2,3}=0,x_4=2$;     (b)  $P_1(-\sqrt{2},-4),P_2(0,0),P_3(\sqrt{2},-4)$
9. The function has the equation $f(x)=x^3+3x^2+3x+2$.
   

   (a)   Find the stationary points of this function.

   (b)   Draw the graph using your GDC and verify the obtained result.

   Solutions:    (a)  $P_1(-1,1)$
10. The function has the equation $f(x)={\displaystyle \frac{x}{2}+\frac{2}{x}}$.
    

    (a)   Find the stationary points of this function.

    (b)   Draw the graph using your GDC and verify the obtained result.

    Solutions:    (a)  $P_1(-2,-2),P_2(2,2)$

Second derivative

11. Write down the first and the second derivative of each of the following functions:
    

    (a)   $f(x)=x^3+2x^2-5x+4$

    (b)   $f(x)=\frac{x^2+3x+1}{5}$

    (c)   $f(x)=\sqrt{x^5}$

    Solutions:    (a)  $f^{\prime }(x)=3x^2+4x-5,f^{″}(x)=6x+4$;     (b)  $f^{\prime }(x)=\frac{2x+3}{5},f^{″}(x)=\frac{2}{5}$;     (c)  $f^{\prime }(x)=\frac{5}{2}\sqrt{x^3},f^{″}(x)=\frac{15}{4}\sqrt{x}$
12. Find $y^{\prime }$ and $y^{″}$ of each of the following functions:
    

    (a)   $y=e^x+\sin x$

    (b)   $y=x-\ln x$

    (c)   $y=x^2\ln x$

    Solutions:    (a)  $y^{\prime }=e^x+\cos x,y^{″}=e^x-\sin x$;     (b)  $y^{\prime }=1-\frac{1}{x},y^{″}=\frac{1}{x^2}$;     (c)  $y^{\prime }=2x\ln x+x,y^{″}=2\ln x+3$
13. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ and $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ of each of the following functions:
    

    (a)   $y=e^x\sin x$

    (b)   $y=\frac{1}{x+3}$

    Solutions:    (a)  $\frac{\mathrm{d}y}{\mathrm{d}x}=e^x\sin x+e^x\cos x,\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=2e^x\cos x$;     (b)  $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{(x+3{)}^2},\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=\frac{2}{(x+3{)}^3}$
14. Given that $a$ and $b$ are constants find the first and the second derivative of each of the following functions:
    

    (a)   $y=x^2+ax+b$

    (b)   $y=\frac{x}{a}+\frac{b}{x}$

    Solutions:    (a)  $y^{\prime }=2x+a,y^{″}=2$;     (b)  $y^{\prime }=\frac{1}{a}-\frac{b}{x^2},y^{″}=\frac{2b}{x^3}$
15. Find the first, the second … and the $n$-th derivative of the function $f(x)=x{e}^x$.
    Solutions:    $f^{\prime }(x)=(x+1)e^x,f^{″}(x)=(x+2)e^x,\dots ,f^{(n)}(x)=(x+n)e^x$

Determining the type of stationary points

16. The function has the equation:  $f(x)=x^3-3x^2+1$.
    

    (a)   Using the first derivative find the stationary points.

    (b)   Using the second derivative determine the type of each of these points.

    Solutions:    Maximum: $P_1(0,1)$,  minimum: $P_2(2,-3)$
17. The function has the equation:  $f(x)=x^4-4x^2$.
    

    (a)   Find the stationary points.

    (b)   Determine the nature of each stationary point.

    Solutions:    Minimum: $P_1(-\sqrt{2},-4)$,  maximum: $P_2(0,0)$,  minimum: $P_3(\sqrt{2},-4)$
18. The function has the equation:  $f(x)=6x^4+8x^3+3$.
    

    (a)   Find the stationary points.

    (b)   Determine the nature of each stationary point.

    (c)   Using your GDC draw the graph to verify the obtained results.

    Solutions:    Minimum: $P_1(-1,1)$,  inflexion point: $P_2(0,3)$
19. The function has the equation:  $f(x)=\frac{2x-1}{x^2}$.
    

    (a)   Find the zeros, vertical asymptotes and horizontal asymptote.

    (b)   Find the stationary points and determine their types.

    (c)   Hence draw the graph.

    Solutions:    (a)  Zeros: $x_1=\frac{1}{2}$, vertical asymptote: $x=0$, horizontal asymptote: $y=0$;     (b)  maximum: $P_1(1,1)$
20. The function has the equation:  $f(x)=\frac{x^2-3x}{x^2+3}$.
    

    (a)   Find the zeros, vertical asymptotes and horizontal asymptote.

    (b)   Find and classify the stationary points.

    (c)   Hence draw the graph.

    Solutions:    (a)  Zeros: $x_1=0,x_2=3$, no vertical asymptotes, horizontal asymptote: $y=1$;     (b)  maximum: $P_1(-3,\frac{3}{2})$, minimum: $P_2(1,-\frac{1}{2})$
21. The function has the equation:  $f(x)=\frac{6}{x^2-4x+6}$.
    

    (a)   Find the zeros and asymptotes, if any.

    (b)   Find and classify the stationary points.

    (c)   Hence draw the graph.

    Solutions:    (a)  No zeros, no vertical asymptotes, horizontal asymptote: $y=0$;     (b)  maximum: $P_1(2,3)$
22. The function has the equation:  $f(x)=\frac{1+\ln x}{x}$.
    

    (a)   Find and classify the stationary points.

    (b)   Using your GDC draw the graph to verify the obtained results.

    Solutions:    (a)  Maximum: $P_1(1,1)$

Different types of inflexion points

23. Find the points where $y^{″}=0$. Use your GDC to draw the graphs and investigate the behaviour of the functions.
    

    (a)   $y=x^3-3x^2+3x$

    (b)   $y=x^3-3x^2+2x+1$

    (c)   $y=x^3-3x^2+4x-1$

    Solutions:    In all three cases $y^{″}=0$ at $P_1(1,1)$. These points are the inflexion points.
    In case (a) this point is also a stationary point  –  it's a stationary inflexion point.
    In cases (b) and (c) this point is not a stationary point  –  it's a non-stationary inflexion point.
24. The function has the equation $y=x^3-6x^2+9x$.
    

    (a)   Find zeros.

    (b)   Find and classify the stationary points.

    (c)   Find and classify the inflexion points.

    (d)   Hence draw the graph.

    Solutions:    (a)  Zeros: $x_1=0,x_{2,3}=3$;     (b)  maximum: $P_1(1,4)$, minimum: $P_2(3,0)$;     (c)  non-stationary inflexion point: $P_3(2,2)$
25. The function has the equation $y=\frac{1}{9}{x}^4+\frac{4}{9}{x}^3$.
    

    (a)   Find zeros.

    (b)   Find and classify the stationary points.

    (c)   Find and classify the inflexion points.

    (d)   Hence draw the graph.

    Solutions:    (a)  Zeros: $x_{1,2,3}=0,x_4=-4$;     (b,c)  minimum: $P_1(-3,-3)$, stationary inflexion point: $P_2(0,0)$, non-stationary inflexion point: $P_3(-2,-\frac{16}{9})$
26. The function has the equation $f(x)=\frac{12}{x^2+3}$.
    

    (a)   Find the inflexion points.

    (b)   Write the equation of the tangent at the inflexion point with the negative abscissa.

    Solutions:    (a)  Inflexion points (both non-stationary): $P_1(-1,3),P_2(1,3)$;     (b)  tangent: $y=\frac{3}{2}x+\frac{9}{2}$

 Index