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)  \(f'(x)=3x^2-6x+4\);     (b)  \(f'(x)=4x^3+3x^2-2x\);     (c)  \(f'(x)=18x+12\)
2. Find derivatives of the following functions:
   

   (a)   \(f(x)=x^2+\frac{\textstyle 1}{\textstyle x}\)

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

   (c)   \(f(x)=\frac{\textstyle x^2+3x+1}{\textstyle x}\)

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

   (a)   \(y=\sqrt{x}\)

   (b)   \(y=\sqrt[\scriptstyle 3]{x}+\sqrt[\scriptstyle 4]{x}\)

   (c)   \(y=\frac{\textstyle 2}{\textstyle \sqrt{x^3}}\)

   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 \(y'\) 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)  \(y'=\cos x-3\sin x\);     (b)  \(y'=2x+e^x\);     (c)  \(y'=\frac{1}{x}+1\)

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)  \(f'(x)=3x^2\sin x+x^3\cos x\);     (b)  \(f'(x)=e^x(x^2+x+1)+e^x(2x+1)=e^x(x^2+3x+2)\);     (c)  \(f'(x)=1+1\cdot\ln x+x\cdot\frac{1}{x}=2+\ln x\)
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)  \(f'(x)=\frac{3}{x^2+6x+9}\);     (b)  \(f'(x)=\frac{-x^2-4x+1}{x^4+2x^2+1}\);     (c)  \(f'(x)=-\frac{2x}{x^4+8x^2+16}\)
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)  \(f'(x)=\frac{xe^x}{x^2+2x+1}\);     (b)  \(f'(x)=\frac{1}{\cos^2 x}\);     (c)  \(f'(x)=\frac{\cos x+2}{4\cos^2x+4\cos x+1}\)
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 \(f(x)=\frac{\textstyle e^x}{\textstyle x-1}\) at \(x=0\).
    Solution:    Tangent: \(y=-2x-1\)
25. Write the equation of the tangent to the graph of the function \(f(x)=\frac{\textstyle x\ln x}{\textstyle 2}\) 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 \(f(x)=\frac{\textstyle 2x-3}{\textstyle 3x+1}\) 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 \(y=\frac{\textstyle 2x}{\textstyle 2-x^2}\) 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)  \(f'(x)=\frac{2}{\sqrt{4x+1}}\);     (b)  \(f'(x)=-4\sin(4x+\pi)\);     (c)  \(f'(x)=\frac{3}{3x+2}\);     (d)  \(f'(x)=6(2x+3)^2\)
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)  \(f'(x)=\frac{x}{\sqrt{x^2+9}}\);     (b)  \(f'(x)=-2xe^{-x^2}\);     (c)  \(f'(x)=\frac{2x+1}{x^2+x+1}\)
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 \(y=\frac{\textstyle 1}{\textstyle \sqrt[\scriptstyle3]{x-9}}\) 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 \(f(x)=\sqrt[\scriptstyle3]{2x-5}\). 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:    \(\theta\doteq78^\circ7'\)
37. Find the angle of inclination of the curve \(y=\frac{\textstyle x}{\textstyle x+2}\) at \(x=1\). Write the angle in degrees and minutes.
    Solution:    \(\theta\doteq12^\circ32'\)
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)  \(\theta\doteq63^\circ26'\)
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)  \(\theta_1\doteq14^\circ2',~ \theta_2\doteq78^\circ41'\);     (c)  \(\alpha=64^\circ39'\)
40. The functions have the equations \(f(x)=2x^2\) and \(g(x)=x-2+\frac{\textstyle 1}{\textstyle x}\).
    

    (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)  \(\theta_1\doteq63^\circ26',~ \theta_2\doteq108^\circ26'\);     (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