1 Table of derivatives

Table 1 lists some of the common functions used in engineering and their corresponding derivatives. Remember that in each case the function in the right-hand column gives the rate of change, or the gradient of the graph, of the function on the left at a particular value of $x$ .

N.B. The angle must always be in radians when differentiating trigonometric functions.

For example, if $y=x^3$ then $\frac{dy}{dx}=3x^2$ .

Example 2

Use Table 1 to find $\frac{dy}{dx}$ when $y$ is given by

1.   $7x$   
2. 14  
3. $5x^2$   
4.   $4x^7$

Solution

1. We note that $7x$ is of the form $kx$ where $k=7$ . Using Table 1 we then have $\frac{dy}{dx}=7$ .
2. Noting that 14 is a constant we see that $\frac{dy}{dx}=0$ .
3. We see that $5x^2$ is of the form $k{x}^n$ , with $k=5$ and $n=2$ . The derivative, $kn{x}^{n-1}$ , is then

     $10x^1$ , or more simply, $10x$ . So if $y=5x^2$ , then $\frac{dy}{dx}=10x$ .

4. We see that $4x^7$ is of the form $k{x}^n$ , with $k=4$ and $n=7$ . Hence the derivative, $\frac{dy}{dx}$ , is

    given by $28x^6$ .

Task!

Use Table 1 to find $\frac{dy}{dx}$ when $y$ is

1.   $\sqrt{x}$
2.   $\frac{5}{x^3}$

1. Write $\sqrt{x}$ as $x^{\frac{1}{2}}$ , and use the result for differentiating $x^n$ with $n=\frac{1}{2}$ .

   Answer

   $\frac{dy}{dx}=n{x}^{n-1}=\frac{1}{2}{x}^{\frac{1}{2}-1}=\frac{1}{2}{x}^{-\frac{1}{2}}$ . This may be written as $\frac{1}{2\sqrt{x}}$ .

2. Write $\frac{5}{x^3}$ as $5x^{-3}$ and use the result for differentiating $k{x}^n$ with $k=5$ and $n=-3$ .

   Answer

   $5\left(-3\right)x^{-3-1}=-15x^{-4}$ Although Table 1 is written using $x$ as the independent variable, the Table can be used for any variable.

Task!

Use Table 1 to find

1. $\frac{dz}{dt}$ given  $z=e^t$   
2. $\frac{dp}{dt}$ given $p=e^{8t}$   
3. $\frac{dz}{dy}$ given $z=e^{-3y}$

1. Answer

   From Table 1, if $y=e^x$ , then $\frac{dy}{dx}=e^x$ . Hence if $z=e^t$ then $\frac{dz}{dt}=e^t$ .

2. Answer

   $8e^{8t}$

3. Answer

   $-3e^{-3y}$

Task!

Find the derivative, $\frac{dy}{dx}$ , when $y$ is

1. $sin2x$   
2. $cos\frac{x}{2}$   
3. $tan5x$

1. Use the result for $sinkx$ in Table 1, taking $k=2$ :

   Answer

   $2cos2x$

2. Note that $cos\frac{x}{2}$ is the same as $cos\frac{1}{2}x$ . Use the result for $coskx$ in Table 1:

   Answer

   $-\frac{1}{2}sin\frac{x}{2}$

3. Use the result for $tankx$ in Table 1:

   Answer

   $5{sec}^{2}5x$

Exercises

1. Find the derivatives of the following functions with respect to $x$ :
   1. $9x^2$   
   2. $5$   
   3. $6x^3$   
   4. $-13x^4$
2. Find $\frac{dz}{dt}$ when $z$ is given by:
   1. $\frac{5}{t^3}$   
   2. $\sqrt{t^3}$   
   3. $5t^{-2}$   
   4. $-\frac{3}{2}{t}^{\frac{3}{2}}$   
   5. $ln5t$
3. Find the derivative of each of the following with respect to the appropriate variable:
   1. $sin5x$   
   2. $cos4t$   
   3. $tan3r$   
   4. $e^{2v}$   
   5. $\frac{1}{e^{3t}}$
4. Find the derivatives of the following with respect to $x$ :
   1. $cos\frac{2x}{3}$   
   2. $sin\left(-2x\right)$   
   3. $tan\pi x$   
   4. $e^{\frac{x}{2}}$   
   5. $ln\frac{2}{3}x$

Answer