Also, which is the instantaneous velocity of the body at time ‘t’.
Example 10
Find the gradient of the curve y = 2 x^{2}
 6 x at (2 , 4 ) from the first principle.
1) y = f (x) = 2 x^{2} 
6 x
2) y + D
y = f ( x + D
x ) = 2 ( x + D
x ) ^{2} 
6 ( x + D x )
Subtracting
D y = 2
x^{2 } + 4 x D
x + 2 ( D
x )^{2 } 
6 x  6
D x  2 x^{2 } + 6 x
= 4 x D
x + 2 (D
x)^{2 } 
6 D x
3) \
= 4 x + 2 D
x  6
4) \
= ( 4 x
+ 2 D x  6 )
= 4 x + 2 (
0 )  6
= 2
\ The gradient at ( 2 , 4 ) is 2.
Example 11
Differentiate y = Öx w . r. to
x, from the first principle.
1) y = Öx
\ y +
D y =
2) Subtracting
