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

Question

fund derivative using limit definition

Answer 1

Let y=f(x).

Let y+k=f(x+h) such that h and k are small enough to discount any 2nd degree order or higher.

y+k=7(x+h)⁵-5(x+h)³=7(x⁵+5hx⁴)-5(x³+3hx²), because we ignore terms containing h², k², and higher powers.

y+k=7x⁵+35hx⁴-5x³-15hx². y=7x⁵-5x³.

Substituting for y: k=35hx⁴-15hx². Therefore k/h=35x⁴-15x².

As h→0, k→0, and h/k→dy/dx, the derivative=35x⁴-15x².