Differentiation

Finding the derivative

1. ?
   ?

   Derivative of a function tells us how steep is this function at a given point.
   Derivative of the function $f(x)=x^n$ is:
   
   $f^{\prime }(x)=n{x}^{n-1}$
   
   Derivative can be written as $f^{\prime }(x)$  or  $y^{\prime }$  or  $\frac{\mathrm{d}y}{\mathrm{d}x}$. The procedure of finding a derivative is called differentiation.
   Find derivatives of the following functions:
   

   (a)   $f(x)=x^3-3x^2+4x+1$

   (b)   $f(x)=x^4+2x^3-5x$

   (c)   $f(x)=-2x^2+5x-13$

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

   (a)   $f(x)=\frac{1}{2}{x}^2+3x-7$

   (b)   $f(x)=x^2(x^2+x-1)$

   (c)   $f(x)=(3x+2{)}^2$

   Solutions:    (a)  $f^{\prime }(x)=x+3$;     (b)  $f^{\prime }(x)=4x^3+3x^2-2x$;     (c)  $f^{\prime }(x)=18x+12$
3. Find $y^{\prime }$ 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)  $f^{\prime }(x)=1-\frac{1}{x^2}$
4. 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)  $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{3}{\sqrt{x^5}}$
5. ?
   ?

   Derivatives of functions $e^x$ and $\ln x$ are:
   
   $(e^x{)}^{\prime }=e^x$
   
   $(\ln x{)}^{\prime }=\frac{1}{x}$
   Find derivatives of the following functions:
   

   (a)   $f(x)=2x+e^x$

   (b)   $f(x)=5e^x+x^2$

   (c)   $f(x)=x^3+\ln x$

   (d)   $f(x)=3e^x+2\ln x$

   Solutions:    (a)  $f^{\prime }(x)=2+e^x$;     (b)  $f^{\prime }(x)=5e^x+2x$;     (c)  $f^{\prime }(x)=3x^2+\frac{1}{x}$;     (d)  $f^{\prime }(x)=3e^x+\frac{2}{x}$

Tangents and normals

6. ?
   ?

   Derivative of a function is equal to the gradient of the tangent at a given point:
   
   $m=f^{\prime }(x_1)$
   
   Equation of the tangent can be found using the form:
   
   $y=mx+c$    or    $y-y_1=m(x-x_1)$
   Consider the function $f(x)=x^3-x+1$ and point $P(1,1)$ which lies on the graph of this function.
   

   (a)   Find the derivative $f^{\prime }(x)$.

   (b)   Calculate the gradient of the tangent.

   (c)   Write down the equation of the tangent.

   Solutions:    (a)  $f^{\prime }(x)=3x^2-1$;     (b)  $m=f^{\prime }(1)=2$;     (c)  $y=2x-1$
7. 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)  $y=1$
8. Find the equation of the tangent to the graph of the function $f(x)=\frac{1}{3}{x}^2+x-2$ at the point $P(3,y)$.
   Solution:    Tangent: $y=3x-5$
9. Find the equation of the tangent to the graph of the function $y=x^3+2x^2$ at the point $x=1$.
   Solution:    Tangent: $y=7x-4$
10. Find the equation of the tangent to the graph of the function $y=x-x^{-1}+1$ at the point $x=-1$.
    Solution:    Tangent: $y=2x+3$
11. Find the equation of the tangent to the graph of the function $y=\sqrt{x}$ at the point $x=9$.
    Solution:    Tangent: $y=\frac{1}{6}x+\frac{3}{2}$
12. 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: $m_1=m_2=3$

13. Find the equation of the tangent to the graph of the function $f(x)=\sqrt{2x+3}$ at $x=-1$.
    Solution:    Tangent: $y=x+2$
14. Write the equation of the tangent to the graph of the function $f(x)=(x+2)\sqrt{x}$ at $x=4$.
    Solution:    Tangent: $y=\frac{7}{2}x-2$
15. Find the equation of the tangent to the graph of the function ${\displaystyle f(x)=\frac{2}{x-1}}$ at $x=2$.
    Solution:    Tangent: $y=-2x+6$
16. Write the equation of the tangent to the graph of the function ${\displaystyle f(x)=\frac{2x-3}{3x+1}}$ at the point where the graph intercepts the ordinate axis.
    Solution:    Tangent: $y=11x-3$
17. Write the equation of the tangent to the graph of the function ${\displaystyle f(x)=\frac{e^x}{x-1}}$ at $x=0$.
    Solution:    Tangent: $y=-2x-1$
18. Find the equation of the tangent to the graph of the function ${\displaystyle y=\frac{1+e^x}{1-x}}$ at $x=0$.
    Solution:    Tangent: $y=3x+2$
19. 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)  $y=\frac{2}{9}x$
20. 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: $y=\frac{1}{2}x-\frac{1}{2}$
21. Write the equation of the tangent to the graph of function $f(x)=\ln (x^2-3)$ at point $P(2,y)$.
    Solution:    Tangent: $y=4x-8$
22. ?
    ?

    Normal is perpendicular to the graph.
    The gradient of the normal is:
    
    ${\displaystyle m_n=-\frac{1}{m_t}}$
    Consider the function $f(x)=x^3-x+1$.
    

    (a)   Draw the graph of this function.

    (b)   Find the equation of the tangent at $x=1$.

    (c)   Find the equation of the normal at $x=1$.

    Solutions:    (b)  Tangent $y=2x-1$;     (c)  normal: $y=-\frac{1}{2}x+\frac{3}{2}$
23. Write the equation of the normal to the curve ${\displaystyle y=\frac{2x}{2-x^2}}$ at the point $P(2,y)$.
    Solution:    Normal: $y=-\frac{1}{3}x-\frac{4}{3}$
24. Find the equation of the normal to the graph of the function $f(x)=\ln (x-3)$ at $x=4$.
    Solution:    $y=-x+4$
25. Find the equation of the normal to the graph of the function $f(x)=x\sqrt{x+4}$ at $x=-3$.
    Solution:    $y=2x+3$
26. 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:    $y=-\frac{1}{2}x+1$
27. Write the equation of the normal to the curve ${\displaystyle y=\frac{1}{\sqrt[3]{x-9}}}$ at the point $P_0(8,y_0)$.
    Solution:    Normal: $y=3x-25$
28. 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)  $B(-3,-7)$
29. Consider the function ${\displaystyle y=\frac{x}{x+2}}$
    

    (a)    Draw the graph of this function.

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

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

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

    Solutions:    (b)  $y=-2x$;     (c)  $y=-\frac{1}{2}x+\frac{3}{2}$;     (d)  $P(-1,2)$
30. The function has the equation:  $f(x)=\sqrt{2x+7}$.
    

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

    This normal and both coordinate axes form a triangle.

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

    (c)   Calculate the area of this triangle.

    Solutions:    (a)  normal: $y=-3x+6$;     (b)  vertices: $P_1(0,0),P_2(2,0),P_3(0,6)$;     (c)  area: $A=6$

Finding points with given gradient

31. Consider the function $f(x)=x^2-3x-2$.

    (a)   Find the derivative $f^{\prime }(x)$.

    (b)   Hence, find the point on the graph where the gradient is 1.

    Solutions:    (a)  $f^{\prime }(x)=2x-3$;     (b)  $P(2,-4)$
32. 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 $P_2(-2,7)$.
33. The function has the equation $f(x)=x^3-6x^2+9x+1$. Find the points on the graph where the gradient is 0. Write down the coordinates of these points.
    Solutions:    $P_1(1,5),P_2(3,1)$
34. The function has the equation $f(x)=\frac{1}{3}{x}^3-\frac{1}{2}{x}^2+1$. Find the points on the graph where the tangent is parallel to the line $y=2x$. Write down the coordinates of these points.
    Solutions:    $P_1(-1,\frac{1}{6}),P_2(2,\frac{5}{3})$
35. The function has the equation $f(x)=x^3-3x^2$. Find the point on the graph where the tangent is parallel to the line $y=-3x$. Write down the coordinates of this point.
    Solution:    $P(1,-2)$

 Index