Example : 12 Find the derivative of sin x wrt x from first principles. Solution Let f(x) = sin x Using the definition of derivative, h⎞ h ⎛ h 2 cos⎜ x + ⎟ sin sin 2⎠ 2 sin( x + h ) − sin x ⎝ 2 f′(x) = hlim = hlim = cos x . hlim = cos x →0 → 0 → 0 h h h 2 2 2 sin θ ⎛ ⎞ = 1⎟ ⎜ u sin g lim θ → 0 θ ⎝ ⎠
Hence f′(x) = cos x Example : 13 Differentiate ax wrt x from first principles Solution Let f(x) = ax f ( x + h) − f ( x ) Using the definition of derivatives f′(x) = hlim →0 h
⇒
h a x +h − a x x lim a − 1 = ax log a f′(x) = hlim = a . →0 h→0 h h
t ⎛ ⎞ ⎜ u sin g lim a − 1 = log a ⎟ ⎜ ⎟ t t →0 ⎝ ⎠
Hence f′(x) = ax log a Example : 14 Differentiate sin (log x) wrt x from first principles Solution Let f(x) = sin (log x) Using the definition of derivatives f ( x + h) − f ( x ) f′(x) = hlim →0 h sin log( x + h) − sin log x f′(x) = hlim = hlim →0 →0 h
⎛ log( x + h) + log x ⎞ ⎛ log( x + h) − log x ⎞ 2 cos⎜ ⎟ sin ⎜ ⎟ 2 2 ⎝ ⎠ ⎝ ⎠ h
⎛ ⎛ log( x + h) − log x ⎞ ⎞ ⎜ sin⎜ ⎟⎟ 2 ⎜ ⎝ ⎠⎟ ⎛ log( x + h) + log x ⎞ ⎟ ⎜ ⎟ × lim ⎜ = hlim 2 cos h →0 h →0 ⎜ 2 ⎝ ⎠ ⎟⎟ ⎜ ⎠ ⎝ ⎛ log( x + h) − log x ⎞ sin⎜ ⎟ 2 ⎝ ⎠ log( x + h) − log x = 2 cos log x hlim × hlim log( x + h) − log x →0 →0 2h 2 log(1 + h / x ) = 2 cos log x . 1 . hlim →0 2h
sin θ ⎤ ⎡ = 1⎥ ⎢Q θlim → 0 θ ⎦ ⎣
log(1 + h / x ) 1 cos log x = cos log x . hlim . = →0 h/ x x x
log(1 + t ) ⎤ ⎡ = 1⎥ ⎢Q tlim → 0 t ⎦ ⎣
Page # 6.