Understanding Engineering Mathematics

y = f(x)

y + dy = f(x + dx)

dy = f(x + dx) − f(x)

x

dx

x

y

y = f(x)

A

B

Figure 8.2

The gradient (210

➤

) of the extended chordABis then given by

GradientAB=

f(x+δx)−f(x)

δx

AsAgets closer toBthis gradient will get closer to the gradient of the tangent atA,i.e.
closer to the gradient of the curve atA,so:

Gradient of curve atA=lim

δx→ 0

[

f(x+δx)−f(x)

δx

]

where limδx→ 0 means letδxtend to zero in the expression in square brackets.
Note that in all this discussion we are assuming that the curvey=f(x)is continuous
and smooth (i.e. has no breaks or sharp corners).
Taking limits is a sophisticated operation about which we will say more in Chapter 14
(see Section 14.5). For now we will only use very simple ideas.
We can repeat the above discussion at each point, i.e. each value ofx, so the gradient
is afunction ofxcalled thederivativeoff(x)(sometimesdifferential coefficient).
The standard forms of notation for the derivative of a functiony=f(x)with respect
toxare:

dy

dx

=

df(x)

dx

=f′(x)= lim

δx→ 0

[

f(x+δx)−f(x)

δx

]

The process of obtainingdy/dxis calleddifferentiation with respect tox.f′(a)means
df/dxevaluated atx=a.
Evaluation of the derivative by calculating the limit is calleddifferentiation from
first principles. Using this one can construct atable of standard derivatives of the
elementary functions and a number ofrules of differentiationwith which we may deduce
many further derivatives.
The standard approach to such differentiation from first principles is as follows.
For a functiony=f(x),letxincrease by an ‘infinitesimal’ amountδx. Evaluate the