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Find the derivative $\frac{d}{dx}\left(\frac{\sqrt{x-2}}{x^2\cos\left(x\right)}\right)$

Step-by-step Solution

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Final Answer

$\frac{\frac{x^2\cos\left(x\right)}{2\sqrt{x-2}}+\sqrt{x-2}\left(-2x\cos\left(x\right)+x^2\sin\left(x\right)\right)}{x^{4}\cos\left(x\right)^2}$
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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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

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

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Learn how to solve quotient rule of differentiation problems step by step online. Find the derivative d/dx(((x-2)^1/2)/(x^2cos(x))). 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}}. The power of a product is equal to the product of it's factors raised to the same power. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x^2 and g=\cos\left(x\right). Simplify the product -(\frac{d}{dx}\left(x^2\right)\cos\left(x\right)+x^2\frac{d}{dx}\left(\cos\left(x\right)\right)).

Final Answer

$\frac{\frac{x^2\cos\left(x\right)}{2\sqrt{x-2}}+\sqrt{x-2}\left(-2x\cos\left(x\right)+x^2\sin\left(x\right)\right)}{x^{4}\cos\left(x\right)^2}$

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 ((x-2)^0.5)/x^2cosx using the product ruleFind derivative of ((x-2)^0.5)/x^2cosx using the quotient ruleFind derivative of ((x-2)^0.5)/x^2cosx using logarithmic differentiationFind derivative of ((x-2)^0.5)/x^2cosx using the definition

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Function Plot

Plotting: $\frac{\frac{x^2\cos\left(x\right)}{2\sqrt{x-2}}+\sqrt{x-2}\left(-2x\cos\left(x\right)+x^2\sin\left(x\right)\right)}{x^{4}\cos\left(x\right)^2}$

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a
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x
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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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