Question

Derivative of tan(2x+3) from first principle

Answer 1

Answer:

Step-by-step explanation:

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Answer 2

Given:

The expression tan(2x +3)

To Find:

Derivative of tan(2x+3) from first principle.

Solution:

Using first principle , derivative,

• f'(x) = lim h ->0  $\frac{f(x+h) - f(x)}{h}$
• f'(x) = lim h->0{ tan (2x + 2h + 3) - tan(2x +3) }/ h

Applying tan a = sin a / cos a ;

• f'(x) = lim h->0 {sin( 2x + 2h + 3)/cos(2x+2h+3) - sin(2x+3)/cos(2x+3)} / h

• f'(x) = lim h->0 {sin( 2x + 2h + 3)cos(2x+3)  - cos(2x+2h+3)sin(2x+3)} /hcos(2x+2h+3)cos(2x+3)

sin A cos B - cos A sin B = sin A - B

• f'(x) = lim h->0 {sin( 2x + 2h + 3 - (2x + 3))} /hcos(2x+2h+3)cos(2x+3)

• f'(x) = lim h->0  sin 2h /hcos(2x+2h+3)cos(2x+3)

We know, lim x->0 sinx/x = 1

Applying the limits

• lim h->0 sin 2h/h = lim h->0 2 x sin 2h/2h = 2
• f'(x) =  2 / cos (2x+3)cos(2x+3)
• f'(x) = 2sec²(2x+3)

Derivative of tan(2x+3) from first principle is 2sec²(2x+3).