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Use logarithmic differentiation to find the derivative of
y with respect to the independent variable.
qquad

y=(cos x)^(x)
A) In
cos x-x tan x
C)
ln x(cos x)^(x-1)
B)
(cos x)^(x)(ln cos x+x cot x)
D)
(cos x)^(x)(ln cos x-x tan x)

Use logarithmic differentiation to find the derivative of $y$ with respect to the independent variable. $\qquad$ $\newline$ $\mathrm{y}=(\cos \mathrm{x})^{\mathrm{x}}$ $\newline$ A) In $\cos x-x \tan x$ $\newline$ C) $\ln x(\cos x)^{x-1}$ $\newline$ B) $(\cos x)^{x}(\ln \cos x+x \cot x)$ $\newline$ D) $(\cos x)^{x}(\ln \cos x-x \tan x)$

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Find equations of tangent lines using limits

Full solution

Q. Use logarithmic differentiation to find the derivative of

y

with respect to the independent variable.

\qquad

\newline

\mathrm{y}=(\cos \mathrm{x})^{\mathrm{x}}

\newline

A) In

\cos x-x \tan x

\newline

C)

\ln x(\cos x)^{x-1}

\newline

B)

(\cos x)^{x}(\ln \cos x+x \cot x)

\newline

D)

(\cos x)^{x}(\ln \cos x-x \tan x)

Apply Logarithmic Differentiation: Step $1$ : Apply logarithmic differentiation by taking the natural logarithm of both sides. $\newline$ Let's start by setting $\ln(y) = \ln((\cos x)^x)$ . $\newline$ Using the power rule for logarithms, we get $\ln(y) = x \ln(\cos x)$ .
Differentiate with Respect to x: Step $2$ : Differentiate both sides with respect to $x$ . $\newline$ Differentiating $\ln(y)$ gives $(1/y) \frac{dy}{dx}$ on the left side. $\newline$ Differentiating $x \ln(\cos x)$ using the product rule gives $\ln(\cos x) + x \times (-\sin x / \cos x)$ . $\newline$ This simplifies to $\ln(\cos x) - x \tan x$ .
Solve for $\frac{dy}{dx}$ : Step $3$ : Solve for $\frac{dy}{dx}$ . $\newline$ Multiply both sides by $y$ to isolate $\frac{dy}{dx}$ . $\newline$ $\frac{dy}{dx} = y(\ln(\cos x) - x \tan x)$ . $\newline$ Substitute back for $y = (\cos x)^x$ . $\newline$ $\frac{dy}{dx} = (\cos x)^x(\ln(\cos x) - x \tan x)$ .

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