Questions: Utilizando as regras de derivação, derive as funções: a) S(p) = sqrt(p) - p b) y = 3 e^x + 4/sqrt[3](x) c) g(x) = x sen(x) d) f(x) = Cos(x)/x

Solution Steps

To find the derivatives of the given functions, we will apply the rules of differentiation such as the power rule, product rule, quotient rule, and chain rule as needed.

a) For \( S(p) = \sqrt{p} - p \), use the power rule to differentiate each term separately.

b) For \( y = 3e^x + \frac{4}{\sqrt[3]{x}} \), differentiate the exponential term using the constant multiple rule and the power rule for the second term.

c) For \( g(x) = x \sin(x) \), apply the product rule to differentiate the product of \( x \) and \( \sin(x) \).

To find the derivative of \( S(p) = \sqrt{p} - p \), we apply the power rule. The derivative of \( \sqrt{p} \) is \( \frac{1}{2\sqrt{p}} \) and the derivative of \( -p \) is \(-1\).

Thus, the derivative is: \[ S'(p) = \frac{1}{2\sqrt{p}} - 1 \]

For the function \( y = 3e^x + \frac{4}{x^{1/3}} \), we differentiate each term separately. The derivative of \( 3e^x \) is \( 3e^x \). For \( \frac{4}{x^{1/3}} \), rewrite it as \( 4x^{-1/3} \) and apply the power rule to get \(-\frac{4}{3}x^{-4/3}\).

Thus, the derivative is: \[ y' = 3e^x - \frac{4}{3}x^{-4/3} \]

To differentiate \( g(x) = x \sin(x) \), we use the product rule. The derivative of \( x \) is \( 1 \) and the derivative of \( \sin(x) \) is \( \cos(x) \).

Applying the product rule, we get: \[ g'(x) = x \cos(x) + \sin(x) \]

Final Answer