List of Standard derivatives
𝑦 = 𝑓(𝑥) 𝑑𝑦
𝑑𝑥
1 𝑥𝑛 𝑛𝑥 𝑛−1
2 𝑒𝑥 𝑒𝑥
3 𝑒 𝑘𝑥 𝑘𝑒 𝑘𝑥
4 𝑎𝑥 𝑎 𝑥 . ln 𝑎
5 ln 𝑥 1ൗ
𝑥
6 log 𝑎 𝑥 1ൗ
𝑥. ln 𝑎
7 sin 𝑥 cos 𝑥
8 cos 𝑥 −sin 𝑥
9 tan 𝑥 sec 2 𝑥
10 cot 𝑥 −cosec 2 𝑥
11 sec 𝑥 sec 𝑥 tan 𝑥
12 cosec 𝑥 −cosec 𝑥 cot 𝑥
13 sinh 𝑥 cosh 𝑥
14 cosh 𝑥 sinh 𝑥
, Function of a function(composite rule)
Sin x is a function of x since the value of sin x depends
the value of the angle x. similarly, sin(2x+5) is a function
the angle (2x+5) since the value of the sine depends on
value of this angle.
i.e sin(2x+5) is a function of (2x+5)
But (2x+5) is itself a function of x, since its value depend
x.
i.e (2x+5) is a function of x
Sin(2x+5) is therefore a function of a function of x and s
expressions are referred to generally as functions of a
function.
So 𝑒 𝑠𝑖𝑛𝑦 is a function of a function of y
, Example
1. If 𝑦 = cos 5𝑥 − 4 , differentiate with respect to x
Solution
Let 𝑈 = 5𝑥 − 4 , ∴ 𝑦 = 𝑐𝑜𝑠𝑢
𝑑𝑦
∴ = −𝑠𝑖𝑛𝑢
𝑑𝑥
𝑑𝑦 𝑑𝑦
But this gives us , 𝑛𝑜𝑡 .
𝑑𝑢 𝑑𝑥
07-Jun-20 MTH 121- General Mathematics II (Calculus)
𝑦 = 𝑓(𝑥) 𝑑𝑦
𝑑𝑥
1 𝑥𝑛 𝑛𝑥 𝑛−1
2 𝑒𝑥 𝑒𝑥
3 𝑒 𝑘𝑥 𝑘𝑒 𝑘𝑥
4 𝑎𝑥 𝑎 𝑥 . ln 𝑎
5 ln 𝑥 1ൗ
𝑥
6 log 𝑎 𝑥 1ൗ
𝑥. ln 𝑎
7 sin 𝑥 cos 𝑥
8 cos 𝑥 −sin 𝑥
9 tan 𝑥 sec 2 𝑥
10 cot 𝑥 −cosec 2 𝑥
11 sec 𝑥 sec 𝑥 tan 𝑥
12 cosec 𝑥 −cosec 𝑥 cot 𝑥
13 sinh 𝑥 cosh 𝑥
14 cosh 𝑥 sinh 𝑥
, Function of a function(composite rule)
Sin x is a function of x since the value of sin x depends
the value of the angle x. similarly, sin(2x+5) is a function
the angle (2x+5) since the value of the sine depends on
value of this angle.
i.e sin(2x+5) is a function of (2x+5)
But (2x+5) is itself a function of x, since its value depend
x.
i.e (2x+5) is a function of x
Sin(2x+5) is therefore a function of a function of x and s
expressions are referred to generally as functions of a
function.
So 𝑒 𝑠𝑖𝑛𝑦 is a function of a function of y
, Example
1. If 𝑦 = cos 5𝑥 − 4 , differentiate with respect to x
Solution
Let 𝑈 = 5𝑥 − 4 , ∴ 𝑦 = 𝑐𝑜𝑠𝑢
𝑑𝑦
∴ = −𝑠𝑖𝑛𝑢
𝑑𝑥
𝑑𝑦 𝑑𝑦
But this gives us , 𝑛𝑜𝑡 .
𝑑𝑢 𝑑𝑥
07-Jun-20 MTH 121- General Mathematics II (Calculus)
