Question

Please help me do this mathematics question

Answer 1

`\textbf{(i)}\\\\y=f(x)= (x)/(4+x)\\\\\\(dy)/(dx)= \lim \limits_(h \to 0) (f(x+h)-f(x))/(h)\\\\\\~~~~~=\lim \limits_(h \to 0)((x+h)/(4+x+h)-(x)/(4+x))/(h)\\\\\\~~~~~~=\lim \limits_(h \to 0)\frac{\tfrac{(x+h)(4+x)-x(4+x+h)}{(4+x+h)(4+x)}}{h}\\\\\\~~~~~~= \lim \limits_( h \to 0) (4x+x^2 +4h+hx -4x-x^2 -hx)/(h(4+x+h)(4+x))\\\\\\~~~~~~=\lim \limits_( h \to 0)(4h)/(h(4+x+h)(4+x))\\\\\\~~~~~~=\lim \limits_(h \to 0) (4)/((4+x+h)(4+x))\\`

`~~~~~~~~= ( 4)/((4+x+0)(4+x))\\\\\\~~~~~~~~= (4)/((4+x)^2)`

`\textbf{(ii)}\\\\\text{Given that,}~ y = f(x) = \sin 2x\\\\\\(dy)/(dx) = \lim \limits_(h \to 0) (f(x+h) - f(x) )/(h)\\\\\\~~~~~~=\lim \limits_(h \to 0) (\sin(2x+2h)-\sin 2x)/(h)\\\\\\~~~~~~=\lim \limits_(h \to 0) (2 \sin \left( (2x +2h -2x)/(2) \right) \cos \left((2x+2h+2x)/(2) \right))/(h)\\\\\\~~~~~~=\lim \limits_(h \to 0) \frac{2 \sin \left( \frac {2h}2 \right) \cos \left(\frac{ 4x+2h} 2\right)}{h}\\\\\\`

`~~~~~~=\lim \limits_(h \to 0) (2 \sin h \cdot \cos \left[ (2(2x+h))/(2)\right] )/(h)\\\\\\~~~~~~=\lim \limits_(h \to 0) (2 \sin h \cdot \cos(2x+h) )/(h)\\\\\\~~~~~~=2\left(\lim \limits_(h \to 0) (\sin h)/( h) \right) \cdot \lim \limits_(h \to 0) \cos (2x+h)\\\\\\~~~~~~=2\cdot 1 \cdot \cos(2x+ 0)\\\\\\~~~~~~=2 \cos 2x`