(d)/(dx)((sin(x))/(cos(x)))=

Identify Function: We need to find the derivative of the function (sin(x))/(cos(x)) with respect to x. This is a quotient, so we will use the quotient rule for differentiation. The quotient rule states that if we have a function h(x) = f(x)/g(x), then its derivative h'(x) is given by (f'(x)g(x) - f(x)g'(x))/(g(x))^2. Here, f(x) = sin(x) and g(x) = cos(x).Apply Quotient Rule: First, we differentiate f(x) = sin(x) with respect to x. The derivative of sin(x) is cos(x). So, f'(x) = cos(x).Differentiate sin(x): Next, we differentiate g(x) = cos(x) with respect to x. The derivative of cos(x) is -sin(x). So, g'(x) = -sin(x).Differentiate cos(x): Now we apply the quotient rule. We have f'(x) = cos(x) and g'(x) = -sin(x), so we plug these into the quotient rule formula: h'(x) = (cos(x) * cos(x) - sin(x) * (-sin(x)))/(cos(x))^2.Plug into Quotient Rule: Simplify the expression in the numerator: cos(x) * cos(x) - sin(x) * (-sin(x)) = cos^2(x) + sin^2(x).Simplify Numerator: We know from the Pythagorean identity that cos^2(x) + sin^2(x) = 1. So, the numerator simplifies to 1.Apply Pythagorean Identity: Now we have h'(x) = 1/(cos(x))^2. We can also write this as h'(x) = sec^2(x), since sec(x) is the reciprocal of cos(x).Final Derivative: Therefore, the derivative of (sin(x))/(cos(x)) with respect to x is sec^2(x).

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$\frac{d}{d x}\left(\frac{\sin (x)}{\cos (x)}\right)=$

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Algebra 2

Simplify variable expressions using properties

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\frac{d}{d x}\left(\frac{\sin (x)}{\cos (x)}\right)=

Identify Function: We need to find the derivative of the function $(\sin(x))/(\cos(x))$ with respect to $x$ . This is a quotient, so we will use the quotient rule for differentiation. The quotient rule states that if we have a function $h(x) = f(x)/g(x)$ , then its derivative $h'(x)$ is given by $(f'(x)g(x) - f(x)g'(x))/(g(x))^2$ . Here, $f(x) = \sin(x)$ and $g(x) = \cos(x)$ .
Apply Quotient Rule: First, we differentiate $f(x) = \sin(x)$ with respect to $x$ . The derivative of $\sin(x)$ is $\cos(x)$ . So, $f'(x) = \cos(x)$ .
Differentiate $\sin(x)$ : Next, we differentiate $g(x) = \cos(x)$ with respect to $x$ . The derivative of $\cos(x)$ is $-\sin(x)$ . So, $g'(x) = -\sin(x)$ .
Differentiate $\cos(x)$ : Now we apply the quotient rule. We have $f'(x) = \cos(x)$ and $g'(x) = -\sin(x)$ , so we plug these into the quotient rule formula: $h'(x) = (\cos(x) \cdot \cos(x) - \sin(x) \cdot (-\sin(x)))/(\cos(x))^2$ .
Plug into Quotient Rule: Simplify the expression in the numerator: $\cos(x) \cdot \cos(x) - \sin(x) \cdot (-\sin(x)) = \cos^2(x) + \sin^2(x)$ .
Simplify Numerator: We know from the Pythagorean identity that $\cos^2(x) + \sin^2(x) = 1$ . So, the numerator simplifies to $1$ .
Apply Pythagorean Identity: Now we have $h'(x) = \frac{1}{(\cos(x))^2}$ . We can also write this as $h'(x) = \sec^2(x)$ , since $\sec(x)$ is the reciprocal of $\cos(x)$ .
Final Derivative: Therefore, the derivative of $\frac{\sin(x)}{\cos(x)}$ with respect to $x$ is $\sec^2(x)$ .