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# Find the derivative $\frac{d}{dt}\left(\frac{\sqrt{t-4}}{t\cos\left(t\right)}\right)$

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##  Final answer to the problem

$\frac{\frac{t\cos\left(t\right)}{2\sqrt{t-4}}+\sqrt{t-4}\left(-\cos\left(t\right)+t\sin\left(t\right)\right)}{t^2\cos\left(t\right)^2}$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Find the derivative using the definition
• Find the derivative using the product rule
• Find the derivative using the quotient rule
• Find the derivative using logarithmic differentiation
• Find the derivative
• Integrate by partial fractions
• Product of Binomials with Common Term
• FOIL Method
• Integrate by substitution
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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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\frac{d}{dt}\left(\sqrt{t-4}\right)t\cos\left(t\right)-\sqrt{t-4}\frac{d}{dt}\left(t\cos\left(t\right)\right)}{\left(t\cos\left(t\right)\right)^2}$

Learn how to solve problems step by step online.

$\frac{\frac{d}{dt}\left(\sqrt{t-4}\right)t\cos\left(t\right)-\sqrt{t-4}\frac{d}{dt}\left(t\cos\left(t\right)\right)}{\left(t\cos\left(t\right)\right)^2}$

Learn how to solve problems step by step online. Find the derivative d/dt(((t-4)^(1/2))/(tcos(t))). 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=. Simplify the product -(\frac{d}{dt}\left(t\right)\cos\left(t\right)+t\frac{d}{dt}\left(\cos\left(t\right)\right)).

##  Final answer to the problem

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

##  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

SnapXam A2

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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

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