Question

Demonstre pela definição que:
a) dx/dx=1
b) dx^2/dx= 2x
c) dx^3/dx=3x^2
d) dx^4/dx=4x^3
e) dx^5/dx=5x^4
De maneira qual
f) dx^h/dx= mx^m-1

Alguém me ajuda, por favor?!

Answer 1

Olá!

A definição é a seguinte:
$\dfrac{\text{d}x^n}{\text{d}x}=n\cdot{x}^{n-1}$

a)
Aqui temos n = 1, visto que o expoente de x é este número.
$\dfrac{\text{d}x^n}{\text{d}x}=n\cdot{x}^{n-1}\\\\\dfrac{\text{d}x^1}{\text{d}x}=1\cdot{x}^{1-1}\\\\\boxed{\dfrac{\text{d}x}{\text{d}x}=x^{0}=1}$


b)
Aqui temos n = 2.
$\dfrac{\text{d}x^n}{\text{d}x}=n\cdot{x}^{n-1}\\\\\dfrac{\text{d}x^2}{\text{d}x}=2\cdot{x}^{2-1}\\\\\boxed{\dfrac{\text{d}x^2}{\text{d}x}=2x}$


c)
Aqui temos n = 3.
$\dfrac{\text{d}x^n}{\text{d}x}=n\cdot{x}^{n-1}\\\\\dfrac{\text{d}x^3}{\text{d}x}=3\cdot{x}^{3-1}\\\\\boxed{\dfrac{\text{d}x^3}{\text{d}x}=3x^2}$


d)
Aqui temos n = 4.
$\dfrac{\text{d}x^n}{\text{d}x}=n\cdot{x}^{n-1}\\\\\dfrac{\text{d}x^4}{\text{d}x}=4\cdot{x}^{4-1}\\\\\boxed{\dfrac{\text{d}x^4}{\text{d}x}=4x^3}$


e)
Aqui temos n = 5.
$\dfrac{\text{d}x^n}{\text{d}x}=n\cdot{x}^{n-1}\\\\\dfrac{\text{d}x^5}{\text{d}x}=5\cdot{x}^{5-1}\\\\\boxed{\dfrac{\text{d}x^5}{\text{d}x}=5x^4}$


f) Aqui temos n = h.
$\dfrac{\text{d}x^n}{\text{d}x}=n\cdot{x}^{n-1}\\\\\dfrac{\text{d}x^h}{\text{d}x}=h\cdot{x}^{h-1}\\\\\boxed{\dfrac{\text{d}x^h}{\text{d}x}=h.x^{h-1}}$


Bons estudos!