Differentiation

Finding the derivative

1. Differentiate the following functions:
   

   (a)   $f(x)=x^3-3x^2+4x+1$

   (b)   $f(x)=x^2(x^2+x-1)$

   (c)   $f(x)=(3x+2{)}^2$

   Solutions:    (a)  $f^{\prime }(x)=3x^2-6x+4$;     (b)  $f^{\prime }(x)=4x^3+3x^2-2x$;     (c)  $f^{\prime }(x)=18x+12$
2. Find derivatives of the following functions:
   

   (a)   $f(x)=x^2+\frac{1}{x}$

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

   (c)   $f(x)=\frac{x^2+3x+1}{x}$

   Solutions:    (a)  $f^{\prime }(x)=2x-\frac{1}{x^2}$;     (b)  $f^{\prime }(x)=-\frac{4}{x^3}-\frac{3}{x^4}$;     (c)  $f^{\prime }(x)=1-\frac{1}{x^2}$
3. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ for the following functions:
   

   (a)   $y=\sqrt{x}$

   (b)   $y=\sqrt[3]{x}+\sqrt[4]{x}$

   (c)   $y=\frac{2}{\sqrt{x^3}}$

   Solutions:    (a)  $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{2\sqrt{x}}$;     (b)  $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{3\sqrt[3]{x^2}}+\frac{1}{4\sqrt[4]{x^3}}$;     (c)  $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{3}{\sqrt{x^5}}$
4. Find $y^{\prime }$ of the following functions:
   

   (a)   $y=\sin x+3\cos x$

   (b)   $y=x^2+e^x$

   (c)   $y=\ln x+x+1$

   Solutions:    (a)  $y^{\prime }=\cos x-3\sin x$;     (b)  $y^{\prime }=2x+e^x$;     (c)  $y^{\prime }=\frac{1}{x}+1$

Tangents and normals

5. Find the equation of the tangent to the curve $y=x^3-3x+3$ at given points:
   

   (a)   $P(0,3)$

   (b)   $P(2,y)$

   (c)   $P(1,y)$

   Solutions:    (a)  $y=-3x+3$;     (b)  $y=9x-13$;     (c)  $y=1$
6. Find the equation of the tangent to the graph of the function $f(x)=\sqrt{x}$ at the point $P(9,y)$.
   Solution:    $y=\frac{1}{6}x+\frac{3}{2}$
7. Find the equation of the tangent to the graph of the function $f(x)=2\cos x$ at $x=\frac{\pi }{2}$.
   Solution:    $y=-2x+\pi$
8. Find the equation of the normal to the graph of the function $f(x)=e^x$ at the point $P$ where $x=0$.
   Solution:    $y=-x+1$
9. Find the equation of the normal to the curve $y=2-\frac{1}{2}x+\sqrt{x}$ at $x=4$.
   Solution:    $y=4x-14$

10. Find the equation of the tangent to the graph of the function $f(x)=\sqrt{2x+3}$ at $x=-1$.
    Solution:    $y=x+2$
11. Find the equation of the tangent to the graph of the function ${\displaystyle f(x)=\frac{2}{x-1}}$ at $x=2$.
    Solution:    $y=-2x+6$
12. Find the equation of the tangent to the graph of the function ${\displaystyle y=\frac{1+e^x}{1-x}}$ at $x=0$.
    Solution:    $y=3x+2$
13. Find the equation of the tangent to the curve ${\displaystyle y=\frac{x-3}{\sqrt{x}}}$
    

    (a)    at $x=1$,

    (b)    at $x=9$.

    Solutions:    (a)  $y=2x-4$;     (b)  $y=\frac{2}{9}x$
14. Find the equation of the normal to the graph of the function $f(x)=\ln (x-3)$ at $x=4$.
    Solution:    $y=-x+4$
15. Find the equation of the normal to the graph of the function $f(x)=x\sqrt{x+4}$ at $x=-3$.
    Solution:    $y=2x+3$
16. Find the equation of the normal to the graph of the function $y=(x-2)\sqrt{x+2}$ at $x$-axis intercept with positive abscissa.
    Solution:    $y=-\frac{1}{2}x+1$
17. Function has the equation $f(x)=x\cos x$
    

    (a)    Write down the equation of the normal at $x=0$.

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

    Solutions:    (a, b)  $y=-x$
18. Function $f$ has the equation ${\displaystyle f(x)=1+\frac{8}{x+2}}$
    

    (a)    Draw the graph of this function.

    (b)    Write down the equation of the normal at point $A(2,y)$.

    (c)    This normal intersects the function in point $A$ and in another point $B$. Find the coordinates of point $B$.

    Solutions:    (b)  $y=2x-1$;     (c)  $B(-3,-7)$
19. Consider the function ${\displaystyle y=\frac{x}{x+2}}$
    

    (a)    Write down the equation of the normal at $x=0$.

    (b)    Write down the equation of the normal at $x=-3$.

    (c)    Find the intersection point of these two normals.

    Solutions:    (a)  $y=-2x$;     (b)  $y=-\frac{1}{2}x+\frac{3}{2}$;     (c)  $P(-1,2)$

Rules of differentiation

20. Differentiate the following functions:
    

    (a)   $f(x)=x^3\sin x$

    (b)   $f(x)=e^x(x^2+x+1)$

    (c)   $f(x)=x+x\ln x$

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

    (a)   ${\displaystyle f(x)=\frac{x}{x+3}}$

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

    (c)   ${\displaystyle f(x)=\frac{1}{x^2+4}}$

    Solutions:    (a)  $f^{\prime }(x)=\frac{3}{x^2+6x+9}$;     (b)  $f^{\prime }(x)=\frac{-x^2-4x+1}{x^4+2x^2+1}$;     (c)  $f^{\prime }(x)=-\frac{2x}{x^4+8x^2+16}$
22. Differentiate the following functions:
    

    (a)   ${\displaystyle f(x)=\frac{e^x}{x+1}}$

    (b)   ${\displaystyle f(x)=\tan x}$

    (c)   ${\displaystyle f(x)=\frac{\sin x}{2\cos x+1}}$

    Solutions:    (a)  $f^{\prime }(x)=\frac{x{e}^x}{x^2+2x+1}$;     (b)  $f^{\prime }(x)=\frac{1}{{\cos }^{2}x}$;     (c)  $f^{\prime }(x)=\frac{\cos x+2}{4{\cos }^{2}x+4\cos x+1}$
23. Write the equation of the tangent to the graph of the function $f(x)=(x+2)\sqrt{x}$ at $x=4$.
    Solution:    Tangent: $y=\frac{7}{2}x-2$
24. Write the equation of the tangent to the graph of the function $f(x)=\frac{e^x}{x-1}$ at $x=0$.
    Solution:    Tangent: $y=-2x-1$
25. Write the equation of the tangent to the graph of the function $f(x)=\frac{x\ln x}{2}$ at the point $P(1,y)$.
    Solution:    Tangent: $y=\frac{1}{2}x-\frac{1}{2}$
26. Write the equation of the tangent to the graph of the function $f(x)=\frac{2x-3}{3x+1}$ at the point where the graph intercepts the ordinate axis.
    Solution:    Tangent: $y=11x-3$
27. Write the equation of the normal to the curve $y=\frac{2x}{2-x^2}$ at the point $P(2,y)$.
    Solution:    Normal: $y=-\frac{1}{3}x-\frac{4}{3}$
28. Differentiate the following functions:
    

    (a)   $f(x)=\sqrt{4x+1}$

    (b)   $f(x)=\cos (4x+\pi )$

    (c)   $f(x)=\ln (3x+2)$

    (d)   $f(x)=(2x+3{)}^3$

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

    (a)   $f(x)=\sqrt{x^2+9}$

    (b)   ${\displaystyle f(x)=e^{-x^2}}$

    (c)   $f(x)=\ln (x^2+x+1)$

    Solutions:    (a)  $f^{\prime }(x)=\frac{x}{\sqrt{x^2+9}}$;     (b)  $f^{\prime }(x)=-2x{e}^{-x^2}$;     (c)  $f^{\prime }(x)=\frac{2x+1}{x^2+x+1}$
30. Write the equation of the tangent to the curve $y=\sqrt{x^2+3}$ at point $P(1,y)$.
    Solution:    Tangent: $y=\frac{1}{2}x+\frac{3}{2}$
31. Write the equation of the tangent to the graph of function $f(x)=\ln (x^2-3)$ at point $P(2,y)$.
    Solution:    Tangent: $y=4x-8$
32. Write the equation of the normal to the curve $y=\frac{1}{\sqrt[3]{x-9}}$ at the point $P_0(8,y_0)$.
    Solution:    Normal: $y=3x-25$
33. The function has the equation $f(x)=x^3-7x+1$. Find the points on the graph where the gradient of this function is 5. Write down the coordinates of these points.
    Solutions:    At points $P_1(2,-5)$ and $P_2(-2,7)$.
34. The gradient of the curve $y=\sqrt{x^2+9}$ at the point $P$ is $\frac{4}{5}$. Find the coordinates of $P$.
    Solution:    $P(4,5)$
35. The function has the equation $f(x)=\sqrt[3]{2x-5}$. Find the points on the graph where the tangent is parallel to the line $2x-3y=0$. Write down the coordinates of these points.
    Solutions:    $P_1(3,1),P_2(2,-1)$

Angle of inclination

36. The function has the equation $f(x)=4x+3\sqrt{x}$. Find the angle of inclination of the tangent to the graph at the point $P(4,y)$. Write the angle in degrees and minutes.
    Solution:    $\theta \doteq {78}^{\circ }{7}^{\prime }$
37. Find the angle of inclination of the curve $y=\frac{x}{x+2}$ at $x=1$. Write the angle in degrees and minutes.
    Solution:    $\theta \doteq {12}^{\circ }{32}^{\prime }$
38. The function has the equation $f(x)=\ln (2x-5)$.
    

    (a)   Find the point where the graph of this function intercepts the $x$ axis.

    (b)   Calculate the angle between the graph and the $x$ axis.

    Solutions:    (a)  $P(3,0)$;     (b)  $\theta \doteq {63}^{\circ }{26}^{\prime }$
39. The functions have the equations $f(x)=\sqrt{x}$ and $g(x)=x^2-3x-2$.
    

    (a)   Show that both graphs pass through the point $P(4,2)$.

    (b)   Find the angles of inclination of the graphs at this point.

    (c)   Calculate the acute angle between the graphs.

    Solutions:    (a)  $f(4)=g(4)=2$;     (b)  ${\theta }_1\doteq {14}^{\circ }{2}^{\prime },{\theta }_2\doteq {78}^{\circ }{41}^{\prime }$;     (c)  $\alpha ={64}^{\circ }{39}^{\prime }$
40. The functions have the equations $f(x)=2x^2$ and $g(x)=x-2+\frac{1}{x}$.
    

    (a)   Find the intersection point of the graphs of these two functions.

    (b)   Find the angles of inclination at this point.

    (c)   Calculate the angle between the graphs.

    Solutions:    (a)  $P(\frac{1}{2},\frac{1}{2})$;     (b)  ${\theta }_1\doteq {63}^{\circ }{26}^{\prime },{\theta }_2\doteq {108}^{\circ }{26}^{\prime }$;     (c)  $\alpha ={45}^{\circ }$
41. The functions have the equations $f(x)=\sqrt{3-x}$ and $g(x)=2x-3$.
    

    (a)   Find the intersection point of the graphs of these two functions.

    (b)   Calculate the angle between the graphs.

    Solutions:    (a)  $P(2,1)$;     (b)  $\alpha ={90}^{\circ }$
42. The function has the equation $f(x)=2x^2-5x+1$. Find the point $P$ on the graph where the angle of inclination is ${45}^{\circ }$. Write down the coordinates of this point.
    Solution:    $P(\frac{3}{2},-2)$
43. The function has the equation $f(x)=\sqrt{2x-1}$. Find the point $P$ on the graph where the angle of inclination is ${30}^{\circ }$. Write down the coordinates of this point.
    Solution:    $P(2,\sqrt{3})$
44. The function has the equation $f(x)=x^3-6x^2+9x+1$. Find the points on the graph where the angle of inclination is $0^{\circ }$. Write down the coordinates of these points.
    Solutions:    $P_1(1,5),P_2(3,1)$

 Index