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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics

Š‡”‡ˆ‘”‡ǡ –Š‡ •Ž‘’‡ ‹• ‰‹˜‡ „›ǣ
∆y
slope = lim - - - - (1.1)
∆x→0 ∆x

The expression in equation (ͳǤͳ) is called the derivative of (y) with respect to (x)
dy
and is represented as: , which is also used to find the slope of the curve at a given point.
dx
dy dy
−
2
dx dx d dy d y
Further, the rate of change of slope: lim 2 1 = = - - - - (1.2)
∆x→0 ∆x dx dx dx 2
dy
is got by appying the concept of limit, to change in , rather than (y).
dx

‘–‡ǣ Š‡ •Ž‘’‡ ‘ˆ –Š‡ †‹•–ƒ ‡ ˜• –‹‡ ‰”ƒ’Š ›‹‡Ž†• ˜‡Ž‘ ‹–›ǡ ƒ† –Š‡ ”ƒ–‡ ‘ˆ Šƒ‰‡ ‘ˆ •Ž‘’‡
›‹‡Ž†• ƒ ‡Ž‡”ƒ–‹‘Ǥ

ͳǤͶǤͳ –ƒ†ƒ”† ‡”‹˜ƒ–‹˜‡•

‘ŽŽ‘™‹‰ ƒ”‡ ƒ ˆ‡™ •–ƒ†ƒ”† †‡”‹˜ƒ–‹˜‡• ‘‘Ž› —•‡† ‹ –Š‡ ‡‰‹‡‡”‹‰ ‘–‡š–Ǥ

ˆȋšȌ ˆ ǯȋšȌ ˆȋšȌ ˆ ǯȋšȌ

-ͳ
•‹ š ‘• š š š
‘• š -•‹ š −2x

a − x 2 2
2
2
–ƒ š •‡ š 2 a − x
ʹ
š š
•‡ š –ƒ š •‡ š ‡ ‡
ƒš
ƒš
‘•‡ š - ‘– š ‘•‡ š ‡ ƒ‡
‘– š - ‘•‡ š Ž‘‰‡ š 1
ʹ
x
cos x
log sin x = cot x
e
sin x

′
f x ∗ g x [Chain Rule] f x ∗ g x + g x ∗ f(x)
′

g(x) g x ∗ f x − f x ∗ g(x)
′
′

f(x) [Division Rule] (f(x)) 2

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
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