(d)/(dx)((e^(x))/(cos(x)))=

Apply Quotient Rule: To find the derivative of the function (e^x)/(cos(x)) with respect to x, we will use the quotient rule. 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) = e^x and g(x) = cos(x).Find Derivative of e^x: First, we find the derivative of f(x) = e^x. The derivative of e^x with respect to x is e^x.Find Derivative of cos(x): Next, we find the derivative of g(x) = cos(x). The derivative of cos(x) with respect to x is -sin(x).Apply Quotient Rule Again: Now we apply the quotient rule. We have f'(x) = e^x and g'(x) = -sin(x), so the derivative of (e^x)/(cos(x)) is (e^x * cos(x) - e^x * (-sin(x)))/(cos(x))^2.Simplify the Expression: Simplify the expression. The derivative is (e^x * cos(x) + e^x * sin(x))/(cos(x))^2.Final Derivative: The final simplified form of the derivative is (e^x * (cos(x) + sin(x)))/(cos(x))^2.

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Algebra 2

Simplify variable expressions using properties

Full solution

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

Apply Quotient Rule: To find the derivative of the function $(e^x)/(\cos(x))$ with respect to $x$ , we will use the quotient rule. 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, $f(x) = e^x$ and $g(x) = \cos(x)$ .
Find Derivative of $\newline$ $e^x$ : First, we find the derivative of $\newline$ $f(x) = e^x$ . The derivative of $\newline$ $e^x$ with respect to $\newline$ $x$ is $\newline$ $e^x$ .
Find Derivative of $\cos(x)$ : Next, we find the derivative of $g(x) = \cos(x)$ . The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$ .
Apply Quotient Rule Again: Now we apply the quotient rule. We have $f'(x) = e^x$ and $g'(x) = -\sin(x)$ , so the derivative of $\frac{e^x}{\cos(x)}$ is $\frac{e^x \cdot \cos(x) - e^x \cdot (-\sin(x))}{(\cos(x))^2}$ .
Simplify the Expression: Simplify the expression. The derivative is $(\frac{e^x \cdot \cos(x) + e^x \cdot \sin(x)}{\cos^2(x)})$ .
Final Derivative: The final simplified form of the derivative is $(e^x \cdot (\cos(x) + \sin(x)))/(\cos(x))^2$ .