Let f(x)=(cos(x))/(x). f^(')(x)=

Apply Quotient Rule: To find the derivative of the function f(x) = (cos(x))/x, we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions, u(x)/v(x), then its derivative f'(x) is given by (v(x)u'(x) - u(x)v'(x))/(v(x))^2. Here, u(x) = cos(x) and v(x) = x.Find u(x): First, we need to find the derivative of u(x) = cos(x). The derivative of cos(x) with respect to x is -sin(x). So, u'(x) = -sin(x).Find v(x): Next, we need to find the derivative of v(x) = x. The derivative of x with respect to x is 1. So, v'(x) = 1.Calculate f'(x): Now we apply the quotient rule. f'(x) = (x * (-sin(x)) - cos(x) * 1) / x^2.Simplify expression: Simplify the expression. f'(x) = (-xsin(x) - cos(x)) / x^2.Further simplify expression: We can further simplify the expression by splitting the fraction. f'(x) = -sin(x)/x - cos(x)/x^2.

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Algebra 2

Simplify variable expressions using properties

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Q. Let

f(x)=\frac{\cos (x)}{x}

\newline

f^{\prime}(x)=

Apply Quotient Rule: To find the derivative of the function $f(x) = \frac{\cos(x)}{x}$ , we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions, $\frac{u(x)}{v(x)}$ , then its derivative $f'(x)$ is given by $\frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = \cos(x)$ and $v(x) = x$ .
Find $u(x)$ : First, we need to find the derivative of $u(x) = \cos(x)$ . The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$ . So, $u'(x) = -\sin(x)$ .
Find $v(x)$ : Next, we need to find the derivative of $v(x) = x$ . The derivative of $x$ with respect to $x$ is $1$ . So, $v'(x) = 1$ .
Calculate $f'(x)$ : Now we apply the quotient rule. $f'(x) = \frac{x \cdot (-\sin(x)) - \cos(x) \cdot 1}{x^2}$ .
Simplify expression: Simplify the expression. $f'(x) = \frac{-x\sin(x) - \cos(x)}{x^2}$ .
Further simplify expression: We can further simplify the expression by splitting the fraction. $f'(x) = -\frac{\sin(x)}{x} - \frac{\cos(x)}{x^2}$ .