Question

prove that d(x^n)/dy=nx^n-1

Answer 1

HELLO DEAR,

YOUR QUESTIONS IS------------>PROVE THAT:- d(xⁿ)/dx = nx^{n - 1}


let y = xⁿ
f(x) = xⁿ
f(x + h) = (x + h)ⁿ

therefore, dy/dx = $\bold{\lim{h\to 0} \frac{f(x + h) - f(x)}{h}}$

dy/dx = $\bold{\lim{h\to 0} \frac{(x + h)^n - x^n}{h}}$

dy/dx = $\bold{\lim{h\to 0} \frac{x^n + ^nC_1x^{n - 1}.h + ^nC_2x^{n - 2} h^2........... - x^n}{h}}$

dy/dx = $\bold{\lim{h \to 0} \frac{^nC_1x^{n - 1}.h + ^nC_2x^{n - 2}.h^2...........}{h}}$

dy/dx = $\bold{\lim{h\to 0} \frac{h(^nC_1x^{n - 1} + ^nC_2x^{n - 2}.h)}{h}}$

dy/dx = $\bold{nx^{n - 1}}$


I HOPE ITS HELP YOU DEAR,
THANKS