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Find the derivative $\frac{d}{dx}\left(\frac{\ln\left(x\cos\left(x\right)\right)}{\sqrt{x+1}}\right)$

Step-by-step Solution

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Final Answer

$\frac{2\left(x+1\right)\left(\cos\left(x\right)-x\sin\left(x\right)\right)-x\ln\left(x\cos\left(x\right)\right)\cos\left(x\right)}{2\sqrt{\left(x+1\right)^{3}}x\cos\left(x\right)}$
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Step-by-step Solution

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Apply the quotient rule for differentiation, which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\sqrt{x+1}\frac{d}{dx}\left(\ln\left(x\cos\left(x\right)\right)\right)-\frac{d}{dx}\left(\sqrt{x+1}\right)\ln\left(x\cos\left(x\right)\right)}{\left(\sqrt{x+1}\right)^2}$

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$\frac{\sqrt{x+1}\frac{d}{dx}\left(\ln\left(x\cos\left(x\right)\right)\right)-\frac{d}{dx}\left(\sqrt{x+1}\right)\ln\left(x\cos\left(x\right)\right)}{\left(\sqrt{x+1}\right)^2}$

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Learn how to solve quotient rule of differentiation problems step by step online. Find the derivative d/dx(ln(xcos(x))/((x+1)^1/2)). Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}. Cancel exponents \frac{1}{2} and 2. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}.

Final Answer

$\frac{2\left(x+1\right)\left(\cos\left(x\right)-x\sin\left(x\right)\right)-x\ln\left(x\cos\left(x\right)\right)\cos\left(x\right)}{2\sqrt{\left(x+1\right)^{3}}x\cos\left(x\right)}$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of lnxcosx/((x+1)^0.5) using the product ruleFind derivative of lnxcosx/((x+1)^0.5) using the quotient ruleFind derivative of lnxcosx/((x+1)^0.5) using logarithmic differentiation

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Function Plot

Plotting: $\frac{2\left(x+1\right)\left(\cos\left(x\right)-x\sin\left(x\right)\right)-x\ln\left(x\cos\left(x\right)\right)\cos\left(x\right)}{2\sqrt{\left(x+1\right)^{3}}x\cos\left(x\right)}$

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0
a
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n
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x
y
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.
(◻)
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◻/◻
/
÷
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e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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