$\left(\frac{d}{dx}\right)(-4x^{\frac{4}{3}}-2x^{-\frac{3}{4}}+3)$

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\left(\frac{d}{dx}\right)(-4x^{\frac{4}{3}}-2x^{-\frac{3}{4}}+3)

Identify Function: Identify the function to differentiate: $f(x) = -4x^{\frac{4}{3}} - 2x^{-\frac{3}{4}} + 3$ . We will use the power rule for differentiation, which states that the derivative of $x^n$ with respect to $x$ is $n\cdot x^{n-1}$ .
Differentiate First Term: Differentiate the first term $-4x^{\frac{4}{3}}$ using the power rule. $\newline$ The derivative of $-4x^{\frac{4}{3}}$ with respect to $x$ is $-4 \times \left(\frac{4}{3}\right) \times x^{\left(\frac{4}{3}-1\right)}$ . $\newline$ Simplify the exponent: $\left(\frac{4}{3}\right) - 1 = \left(\frac{4}{3}\right) - \left(\frac{3}{3}\right) = \frac{1}{3}$ . $\newline$ So, the derivative of the first term is $-4 \times \left(\frac{4}{3}\right) \times x^{\frac{1}{3}}$ .
Differentiate Second Term: Differentiate the second term $-2x^{(-3/4)}$ using the power rule. $\newline$ The derivative of $-2x^{(-3/4)}$ with respect to $x$ is $-2 \times (-3/4) \times x^{((-3/4)-1)}$ . $\newline$ Simplify the exponent: $(-3/4) - 1 = (-3/4) - (4/4) = -7/4$ . $\newline$ So, the derivative of the second term is $-2 \times (-3/4) \times x^{(-7/4)}$ .
Differentiate Third Term: Differentiate the third term $+3$ . The derivative of a constant is $0$ . So, the derivative of the third term is $0$ .
Combine Derivatives: Combine the derivatives of all terms to get the final derivative of the function. $\newline$ The derivative of $f(x)$ is $-4 \times (4/3) \times x^{1/3} + -2 \times (-3/4) \times x^{-7/4} + 0$ . $\newline$ Simplify the coefficients: $-4 \times (4/3) = -16/3$ and $-2 \times (-3/4) = 3/2$ . $\newline$ So, the final derivative of $f(x)$ is $(-16/3)x^{1/3} + (3/2)x^{-7/4}$ .