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Find the derivative of $e=\sqrt{\frac{6^{\left(n+3\right)}6^{\left(n+2\right)}}{\left(\frac{103}{5}\right)^n}}$

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Solution

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Final answer to the problem

$\frac{\ln\left(6\right)6^{\frac{1}{2}\left(2n+5\right)}\left(\frac{103}{5}\right)^{\frac{1}{2}n}-\frac{1}{2}\ln\left(\frac{103}{5}\right)6^{\frac{1}{2}\left(2n+5\right)}\left(\frac{103}{5}\right)^{\frac{1}{2}n}}{\left(\frac{103}{5}\right)^n}$
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Step-by-step Solution

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  • Find the derivative
  • 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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Simplifying

$\frac{d}{dn}\left(\sqrt{\frac{6^{\left(n+3\right)}6^{\left(n+2\right)}}{\left(\frac{103}{5}\right)^n}}\right)$

Learn how to solve differential calculus problems step by step online.

$\frac{d}{dn}\left(\sqrt{\frac{6^{\left(n+3\right)}6^{\left(n+2\right)}}{\left(\frac{103}{5}\right)^n}}\right)$

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Learn how to solve differential calculus problems step by step online. Find the derivative of e=((6^(n+3)6^(n+2))/((103/5)^n))^1/2. Simplifying. Simplifying. The power of a quotient is equal to the quotient of the power of the numerator and denominator: \displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}. 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}}.

Final answer to the problem

$\frac{\ln\left(6\right)6^{\frac{1}{2}\left(2n+5\right)}\left(\frac{103}{5}\right)^{\frac{1}{2}n}-\frac{1}{2}\ln\left(\frac{103}{5}\right)6^{\frac{1}{2}\left(2n+5\right)}\left(\frac{103}{5}\right)^{\frac{1}{2}n}}{\left(\frac{103}{5}\right)^n}$

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Function Plot

Plotting: $\frac{\ln\left(6\right)6^{\frac{1}{2}\left(2n+5\right)}\left(\frac{103}{5}\right)^{\frac{1}{2}n}-\frac{1}{2}\ln\left(\frac{103}{5}\right)6^{\frac{1}{2}\left(2n+5\right)}\left(\frac{103}{5}\right)^{\frac{1}{2}n}}{\left(\frac{103}{5}\right)^n}$

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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
D□x
∫
∫◻
|◻|
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

How to improve your answer:

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Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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