5.1 Differentiation (first principles, rules) and sketching graphs

Differentiation from first principles

The tangent problem has given rise to the branch of calculus called differential calculus and the equation: ${lim}_{h\to 0}\frac{f(x+h)-f\left(x\right)}{h}$ defines the derivative of the function $f\left(x\right)$ . Using [link] to calculate the derivative is called finding the derivative from first principles .

Derivative

The derivative of a function $f\left(x\right)$ is written as $f^{\text{'}}\left(x\right)$ and is defined by:

$$f^{\text{'}}\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$$

There are a few different notations used to refer to derivatives. If we use the traditional notation $y=f\left(x\right)$ to indicate that the dependent variable is $y$ and the independent variable is $x$ , then some common alternative notations for the derivative are as follows: $f^{\text{'}}\left(x\right)=y^{\text{'}}=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}f\left(x\right)=Df\left(x\right)=D_{x}f\left(x\right)$

The symbols $D$ and $\frac{d}{dx}$ are called differential operators because they indicate the operation of differentiation , which is the process of calculating a derivative. It is very important that you learn to identify these different ways of denoting the derivative, and that you are consistent in your usage of them when answering questions.

Derivatives

1. Given $g\left(x\right)=-x^2$
   1. determine ${\displaystyle \frac{g(x+h)-g\left(x\right)}{h}}$
   2. hence, determine ${lim}_{h\to 0}\frac{g(x+h)-g\left(x\right)}{h}$
   3. explain the meaning of your answer in (b).
2. Find the derivative of $f\left(x\right)=-2x^2+3x$ using first principles.
3. Determine the derivative of $f\left(x\right)={\displaystyle \frac{1}{x-2}}$ using first principles.
4. Determine $f^{\text{'}}\left(3\right)$ from first principles if $f\left(x\right)=-5x^2$ .
5. If $h\left(x\right)=4x^2-4x$ , determine $h^{\text{'}}\left(x\right)$ using first principles.

Rules of differentiation

Calculating the derivative of a function from first principles is very long, and it is easy to make mistakes. Fortunately, there are rules which make calculating the derivative simple.

Investigation : rules of differentiation

From first principles, determine the derivatives of the following:

1. $f\left(x\right)=b$
2. $f\left(x\right)=x$
3. $f\left(x\right)=x^2$
4. $f\left(x\right)=x^3$
5. $f\left(x\right)=1/x$

You should have found the following:

If we examine these results we see that there is a pattern, which can be summarised by: $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

There are two other rules which make differentiation simpler. For any two functions $f\left(x\right)$ and $g\left(x\right)$ : $\frac{d}{dx}[f\left(x\right)\pm g\left(x\right)]=f^{\text{'}}\left(x\right)\pm g^{\text{'}}\left(x\right)$ This means that we differentiate each term separately.

The final rule applies to a function $f\left(x\right)$ that is multiplied by a constant $k$ . $\frac{d}{dx}[k.f\left(x\right)]=k{f}^{\text{'}}\left(x\right)$