find the derivative using limit definition f(x)=7x^5-5x^3

Question

find derivative using limit definition

Answer 1

f(x) = 7x^5-5x^3

By the definition

f'(x) = lim(h-->0) f(x+h)-f(x)/h

f(x+h) = 7(x+h)^5-5(x+h)^3

= 7(x^5+5x^4h+10x^3h^2+10x^2h^3+10xh^4+h^5)-5(x^3+3x^2h+3xh^2+h^3)

= 7h^5+35h^4x+70h^3x^2+70h^2x^3+35hx^4+7x^5-5x^3-15x^2h-15xh^2-15xh^2-5h^3

f'(x) = lim(h-->0) f(x+h)-f(x)/h

f'(x) = lim(h-->0) {[7h^5+35h^4x+70h^3x^2+70h^2x^3+35hx^4+7x^5-5x^3-15x^2h-15xh^2-15xh^2-5h^3]-[7x^5-5x^3]}/h

= lim(h-->0)(7h^5+35h^3x+70h^2x^2-5h^2+70hx^3-15hx+35x^4-15x^2

= 35x^4-15x^2