The derivative as a gradient and rate of changeEdexcel International A Level Maths: Revision notes

The gradient of a curve at a point is defined as the gradient of the tangent to the curve at that point: the straight line that just touches the curve there and has the same direction as the curve. For a curve $y=f(x)$ the gradient changes from point to point, so we describe it with a gradient function, the derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$, also written $f'(x)$. Its value at $x=a$ is $f'(a)$, the gradient of the tangent at $x=a$. If $f'(a)>0$ the curve is rising as $x$ increases, if $f'(a)<0$ it is falling, and if $f'(a)=0$ the tangent is horizontal.

A chord joins two points on a curve. For the points with $x$-coordinates $x$ and $x+h$ its gradient is $\frac{f(x+h)-f(x)}{h}.$ This is only an average gradient. As $h$ gets smaller the second point moves towards the first and the chord approaches the tangent. The derivative is defined as the limit $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.$ Using this definition is called differentiation from first principles. Example: $f(x)=x^2$. $\frac{(x+h)^2-x^2}{h}=\frac{2xh+h^2}{h}=2x+h$. As $h\to0$, $f'(x)=2x$.

The derivative of $y$ with respect to $x$ is written $\frac{\mathrm{d}y}{\mathrm{d}x}$ or, if $y=f(x)$, $f'(x)$. It measures the rate of change of $y$ with respect to $x$: how fast $y$ changes per unit increase in $x$. Its units are the units of $y$ divided by the units of $x$. Example: the volume $V$ cm$^3$ of water at time $t$ minutes is $V=40+6t-0.5t^2$. Then $\frac{\mathrm{d}V}{\mathrm{d}t}=6-t$ in cm$^3$ min$^{-1}$. At $t=4$ the volume is increasing at $2$ cm$^3$ min$^{-1}$, and at $t=6$ the rate is zero. A negative rate means the quantity is decreasing.

Differentiating $\frac{\mathrm{d}y}{\mathrm{d}x}$ again gives the second derivative $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=f''(x).$ It is the rate of change of the gradient. Where $f''(x)>0$ the gradient is increasing, and where $f''(x)<0$ it is decreasing. Example: $f(x)=x^3-6x^2+9x+2$ gives $f'(x)=3x^2-12x+9$ and $f''(x)=6x-12$. Since $f''(x)>0$ for $x>2$, the gradient increases for $x>2$. The rule that $x^n$ differentiates to $nx^{n-1}$ is covered in the next subtopic. The chain rule is not needed here.