(d)/(dx)[(x^(3))/(sin(x))]=

Find Derivative of f(x): We need to find the derivative of the function f(x) = (x^3)/(sin(x)) with respect to x. This requires the use of the quotient rule for derivatives, which states that if you have a function that is the quotient of two functions, u(x)/v(x), its derivative is given by (u'(x)v(x) - u(x)v'(x))/(v(x))^2. Here, u(x) = x^3 and v(x) = sin(x).Derivative of u(x): First, we find the derivative of u(x) = x^3 with respect to x. The derivative of x^n with respect to x is n*x^(n-1), so the derivative of x^3 is 3*x^(3-1) = 3x^2.Derivative of v(x): Next, we find the derivative of v(x) = sin(x) with respect to x. The derivative of sin(x) with respect to x is cos(x).Apply Quotient Rule: Now we apply the quotient rule. The derivative of f(x) = (x^3)/(sin(x)) with respect to x is given by (u'(x)v(x) - u(x)v'(x))/(v(x))^2. Substituting the derivatives we found, we get (3x^2*sin(x) - x^3*cos(x))/(sin(x))^2.Simplify Expression: We can simplify the expression by leaving it as it is, since there is no further simplification that combines terms or reduces the expression.

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$\frac{d}{d x}\left[\frac{x^{3}}{\sin (x)}\right]=$

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\frac{d}{d x}\left[\frac{x^{3}}{\sin (x)}\right]=

Find Derivative of $f(x)$ : We need to find the derivative of the function $f(x) = \frac{x^3}{\sin(x)}$ with respect to $x$ . This requires the use of the quotient rule for derivatives, which states that if you have a function that is the quotient of two functions, $\frac{u(x)}{v(x)}$ , its derivative is given by $\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = x^3$ and $v(x) = \sin(x)$ .
Derivative of $u(x)$ : First, we find the derivative of $u(x) = x^3$ with respect to $x$ . The derivative of $x^n$ with respect to $x$ is $n\cdot x^{(n-1)}$ , so the derivative of $x^3$ is $3\cdot x^{(3-1)} = 3x^2$ .
Derivative of $v(x)$ : Next, we find the derivative of $v(x) = \sin(x)$ with respect to $x$ . The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$ .
Apply Quotient Rule: Now we apply the quotient rule. The derivative of $f(x) = \frac{x^3}{\sin(x)}$ with respect to $x$ is given by $\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . Substituting the derivatives we found, we get $\frac{3x^2\sin(x) - x^3\cos(x)}{(\sin(x))^2}$ .
Simplify Expression: We can simplify the expression by leaving it as it is, since there is no further simplification that combines terms or reduces the expression.