## Content

### Definition of the derivative

The method we used in the previous section to find the gradient of a tangent to a graph at a point can actually be used to work out the gradient everywhere, simultaneously.

#### Example

Let $$f(x)=x^2$$. What is the gradient of the tangent line to the graph $$y=f(x)$$ at a general point $$(x,f(x))$$ on this graph?

#### Solution

We calculate the same limit as in previous examples, using the variable $$x$$ in place of a number:

\begin{align*} \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x} &= \lim_{\Delta x \to 0} \dfrac{(x+\Delta x)^2 - x^2}{\Delta x} \\ &= \lim_{\Delta x \to 0} \dfrac{x^2 + 2 x (\Delta x) + (\Delta x)^2 - x^2}{\Delta x} \\ &= \lim_{\Delta x \to 0} \bigl( 2x + \Delta x \bigr) = 2x. \end{align*}

Thus, the gradient of the tangent line at the point $$(x,f(x))$$ is $$2x$$.

The derivative of a function $$f(x)$$ is the function $$f'(x)$$ which gives the gradient of the tangent to the graph $$y=f(x)$$ at each value of $$x$$. It is often also denoted $$\dfrac{dy}{dx}$$. Thus

$f'(x) = \dfrac{dy}{dx} = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x)-f(x)}{\Delta x}.$

The previous example shows that the derivative of $$f(x) = x^2$$ is $$f'(x) = 2x$$.

##### Exercise 2

Show that the derivative of $$f(x) = x^3$$ is $$f'(x) = 3x^2$$.

Functions which have derivatives are called differentiable. Not all functions are differentiable; in particular, to be differentiable, a function must be continuous. Almost all functions we meet in secondary school mathematics are differentiable. In particular, all polynomials, rational functions, exponentials, logarithms and trigonometric functions (such as $$\sin$$, $$\cos$$ and $$\tan$$) are differentiable.

Derivatives of exponential, logarithmic and trigonometric functions are discussed in the two modules Exponential and logarithmic functions and The calculus of trigonometric functions.

We shall say a little more about which functions are differentiable and which are not in the Appendix to this module.

In the following example and exercises, we differentiate constant and linear functions.

##### Exercise 3

Show that the derivative of a constant function $$f(x) = c$$, with $$c$$ a real constant, is given by $$f'(x) = 0$$.

#### Example

What is the derivative of the linear function $$f(x) = 3x + 7$$?

#### Solution

We calculate

\begin{align*} f'(x) &= \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x} \\ &= \lim_{\Delta x \to 0} \dfrac{3(x + \Delta x) + 7 - 3x - 7}{\Delta x} \\ &= \lim_{\Delta x \to 0} \dfrac{3\,\Delta x}{\Delta x} = 3. \end{align*}

Thus $$f'(x) = 3$$. (This answer makes sense, as the graph of $$y=f(x)$$ is the line $$y=3x+7$$, which has gradient 3.)

##### Exercise 4

Show that the derivative of a linear function $$f(x) = ax+b$$, with $$a,b$$ real constants, is $$f'(x) = a$$.

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