Question

A derivada das funções f(x) = 5x2 + 7 e f (x) = square root of x , são respectivamente:

I) 10x e square root of x

II) 5x2 + 7 e bevelled fraction numerator 1 over denominator 2 square root of x end fraction

III) 10x e bevelled fraction numerator 1 over denominator 2 square root of x end fraction

IV) 5x e square root of x

Podemos afirmar que a alternativa correta é:

Escolha uma:
a. I
b. II
c. III
d. IV

Answer 1

Derivada de constante:

$\boxed{\boxed{\dfrac{d}{dx}k=0}}$

Derivada de potências de x:

$\boxed{\boxed{\dfrac{d}{dx}x^{n}=nx^{n-1}~~~~,~~\boxed{n\in\mathbb{R}}}}$

Derivada da soma: Seja f(x) = g(x) + h(x):

$\boxed{\boxed{f'(x)=g'(x)+h'(x)}}$

Da regra do produto ([fg]' = gf' + fg'), tiramos que:

$\boxed{\boxed{\dfrac{d}{dx}(kx^{n})=knx^{n-1}~~~~sendo~k=constante}}$

* Sendo h(x) e g(x) deriváveis
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Derivada de f(x) = 5x² + 7:

$f'(x)=\dfrac{d}{dx}(5x^{2})+\dfrac{d}{dx}7\\\\\\f'(x)=5\dfrac{d}{dx}(x^{2})+0\\\\\\f'(x)=5\cdot2x^{2-1}\\\\\\\boxed{\boxed{f'(x)=10x}}$

Derivada de f(x) = √x:

$f(x)=\sqrt{x}=\sqrt[2]{x^{1}}=x^{1/2}\\\\\\f'(x)=\dfrac{d}{dx}x^{1/2}\\\\\\f'(x)=\dfrac{1}{2}x^{(1/2)-1}\\\\\\f'(x)=\dfrac{1}{2}\cdot x^{(1-2)/2}\\\\\\f'(x)=\dfrac{1}{2}x^{-1/2}\\\\\\f'(x)=\dfrac{1}{2}\cdot\dfrac{1}{x^{1/2}}\\\\\\f'(x)=\dfrac{1}{2}\cdot\dfrac{1}{\sqrt{x}}\\\\\\\boxed{\boxed{f'(x)=\dfrac{1}{2\sqrt{x}}}}$

Resposta: III, letra C