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Function: If the value of a quantity y (say) depends on the value of another quantity x, then y is the function of x i.e. y = f(x).
The quantity y is called dependent variable and the quantity x is called independent variable. For example, y = 2x2 + 4x + 7 is a function of x
(i) When x = 1, y = 2(1)2 + 4x1+7 = 13
(ii) When x = 2, y = 2(2)2 +4x2+7 = 23
As the value of y depends on the value of x, y is the function of x.
Differential coefficient or derivative of a function
Let y = f(x) …. (1)
That is, the value of y depends upon the value of x.
Let ∆x be a small increment in x, so that ∆y is the corresponding small increment in y, then
y + ∆y = f(x+∆x) …. (2)
Subtract (1) from (2), we get ∆y = f(x+∆x) − f(x)
Divide both sides by ∆x
Where is called average rate of change of y w.r.t. x.
Let us ∆x be as small as possible i.e. ∆x→0 (read as delta x tends to zero)
Then differential coefficient or derivative of y w.r.t. x is
Example: Differentiate the following w.r.t. x.
(i) sin 2 x
(ii) x sin x
(i) Let y = sin 2x
(ii) Let y = x sin x
Example: Differentiate the follow w.r.t .x.
(i) 2 (loge x)2
(ii) log(ax + b)
(i) Let y = (loge x)2
(ii) Let y = log(ax + b)
Example: If S = 2t3 - 3t2 + 2,find the position, velocity and acceleration of a particle at the end of 2s. S is measured in metre and t in second
S = 2t3 - 3t2 + 2
When t = 2s, S = 2 × 8 – 3 × 4 + 2 = 6 m