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Find the derivative $\frac{d}{dx}\left(\frac{\ln\left(x^2+3\right)}{5e^x}\right)$

Step-by-step Solution

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Final Answer

$\frac{10xe^x-5e^x\left(x^2+3\right)\ln\left(x^2+3\right)}{25\left(x^2+3\right)e^{2x}}$
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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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{5\frac{d}{dx}\left(\ln\left(x^2+3\right)\right)e^x-\frac{d}{dx}\left(5e^x\right)\ln\left(x^2+3\right)}{\left(5e^x\right)^2}$

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$\frac{5\frac{d}{dx}\left(\ln\left(x^2+3\right)\right)e^x-\frac{d}{dx}\left(5e^x\right)\ln\left(x^2+3\right)}{\left(5e^x\right)^2}$

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Learn how to solve problems step by step online. Find the derivative d/dx(ln(x^2+3)/(5e^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. The derivative of a function multiplied by a constant (5) is equal to the constant times the derivative of the function. 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

$\frac{10xe^x-5e^x\left(x^2+3\right)\ln\left(x^2+3\right)}{25\left(x^2+3\right)e^{2x}}$

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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 ln(x^2+3)/5e^x using the product ruleFind derivative of ln(x^2+3)/5e^x using the quotient ruleFind derivative of ln(x^2+3)/5e^x using logarithmic differentiation

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Function Plot

Plotting: $\frac{10xe^x-5e^x\left(x^2+3\right)\ln\left(x^2+3\right)}{25\left(x^2+3\right)e^{2x}}$

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5
6
7
8
9
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

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