Full solution

3x^{2}z^{2}-6xz^{2}-24z^{2}

Identify Function: Identify the function to differentiate with respect to $x$ : $3x^{2}z^{2}-6xz^{2}-24z^{2}$ . Since $z$ is treated as a constant with respect to $x$ , we will apply the power rule to each term involving $x$ .
Differentiate First Term: Differentiate the first term with respect to $x$ : $\frac{d}{dx}(3x^{2}z^{2})$ . Using the power rule, the derivative of $x^n$ is $nx^{n-1}$ , so the derivative of $3x^{2}z^{2}$ is $2\cdot 3x^{2-1}z^{2} = 6xz^{2}$ .
Differentiate Second Term: Differentiate the second term with respect to $x$ : $\frac{d}{dx}(-6xz^{2})$ . Using the power rule, the derivative of $x$ is $1$ , so the derivative of $-6xz^{2}$ is $-6z^{2}$ .
Differentiate Third Term: Differentiate the third term with respect to $x$ : $\frac{d}{dx}(-24z^{2})$ . Since there is no $x$ in this term, its derivative with respect to $x$ is $0$ .
Combine Derivatives: Combine the derivatives of all terms to get the final derivative of the function with respect to $x$ . $\frac{d}{dx}(3x^{2}z^{2}-6xz^{2}-24z^{2}) = 6xz^{2} - 6z^{2} + 0$ .
Simplify Final Derivative: Simplify the final derivative expression if necessary. $\newline$ The final derivative simplifies to $6xz^{2} - 6z^{2}$ .