Math, asked by sdhingra515, 17 days ago

diffrenciate sec 4X
with the help of 1st principal​

Answers

Answered by Anonymous
12

{\bf f'(x) = \[ \lim_{h\to0}  \dfrac{f(x + h) - f(x)}{h} \]} \\  \\  { \bf \[ \lim_{h\to0}  \dfrac{sec4(x + h) - sec \: x}{h} \]} \\  \\ { \bf \[ \lim_{h\to0}  \dfrac{sec(4x + 4h) - sec \: x}{h} \]} \\  \\ { \bf \[ \lim_{h\to0} \:  \[\left[\frac{1}{h} \left(\dfrac{1}{cos(4x + 4h)}  -  \frac{1}{cos4x} \]\right)\]\right]} \\  \\ { \bf \[ \lim_{h\to0} \:  \[\left[\frac{1}{h} \left(\dfrac{cos \: 4x - cos(4x + 4h)}{cos4x \: . \: cos(4x + 4h)} \]\right)\]\right]} \\  \\ { \boxed{ \red{ \bf{cosA - cosB = - 2 sin\left(\dfrac{A+B}{2}\right)sin\left(\dfrac{A-B}{2}\right)}}}} \\ \\ { \bf \[ \lim_{h\to0} \:  \[\left[\frac{1}{h} \left(\dfrac{ - 2sin \left( \frac{4x + 4x + 4h}{2} \right)sin\left( \frac{4x  -  4x  -  4h}{2} \right)}{cos4x \: . \: cos(4x + 4h)} \]\right)\]\right]} \\  \\ { \bf \[ \lim_{h\to0} \:  \[\left[\frac{1}{h} \left(\dfrac{ - 2sin \left( \frac{8x + 4h}{2} \right)sin\left( \frac{ - 4h}{2} \right)}{cos4x \: . \: cos(4x + 4h)} \]\right)\]\right]} \\  \\ { \bf \[ \lim_{h\to0} \:  \[\left[\frac{1}{h} \left(\dfrac{ - 2sin (4x + 2h)sin(- 2h)}{cos4x \: . \: cos(4x + 4h)} \]\right)\]\right]} \\  \\ { \underline{ \red{ \pmb{ \sf{Divide \: and \: multiply \: by \: 2}}}}} \\  \\ { \bf \[ \lim_{h\to0} \:  \[\left[\dfrac{ - 2.2.sin (4x + 2h)sin(- 2h)}{2.h.cos4x \: . \: cos(4x + 4h)}\]\right]} \\  \\ { \bf \[ \lim_{h\to0} \:  \[\left[\dfrac{ 4sin (4x + 2h)sin \: 2h}{cos4x \: . \: cos(4x + 4h).2h}\]\right]} \\  \\ { \bf \[ \lim_{h\to0} \:  \[\left[\dfrac{sin \: 2h}{2h}\]\right]} \times  { \bf \[ \lim_{h\to0} \: \[\left[\dfrac{ 4sin (4x + 2h)}{cos4x \: . \: cos(4x + 4h)}\]\right]} \\  \\ { \boxed{ \red{ \pmb{ \sf{ \frac{sinx}{x}  = 1}}}}} \\  \\ 1 \times  { \bf \[\left[\dfrac{ 4sin (4x + 0)}{cos4x \: . \: cos(4x + 0)}\]\right]} \\  \\ { \bf \[\left[\dfrac{ 4sin4x}{cos^{2} 4x}\]\right]} \\  \\{ \bf \[\left[\dfrac{ 4sin4x}{cos4x} \times  \frac{1}{cos4x} \]\right]} \\  \\{ \bf 4 \: tan4x.sec4x}


MяƖиνιѕιвʟє: Osmm
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