Question

2. Use the first principles definition to differentiate \( f(x)=2 x^{4}-3 \)

Answer 1

`f'(x)=8x^3`

Explanation:

`\displaystyle f'(x)=\lim_(h\rightarrow0)(f(x+h)-f(x))/(h)\\\\f'(x)=\lim_(h\rightarrow0)(2(x+h)^4-3-(2x^4-3))/(h)\\\\f'(x)=\lim_(h\rightarrow0)(2(x^4+4x^3h+6x^2h^2+4xh^3+h^4)-3-2x^4+3)/(h)\\\\f'(x)=\lim_(h\rightarrow0)(2x^4+8x^3h+12x^2h^2+8xh^3+2h^4-2x^4)/(h)\\\\f'(x)=\lim_(h\rightarrow0)(8x^3h+12x^2h^2+8xh^3+2h^4)/(h)\\\\f'(x)=\lim_(h\rightarrow0)8x^3+12x^2h+8xh^2+2h^3\\\\f'(x)=8x^3+12x^2(0)+8x(0)^2+2(0)^3\\\\f'(x)=8x^3`