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Find the derivative using logarithmic differentiation method $\frac{d}{dx}\left(\left(\sqrt{1+e^{5x}}\right)^{\cos\left(3x\right)}\right)$

Used Formulas

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Solution

Formulas

Videos

Basic Differentiation Rules

$\frac{d}{dx}\left(cx\right)=c\frac{d}{dx}\left(x\right)$
· Product rule for derivatives
$\frac{d}{dx}\left(ab\right)=\frac{d}{dx}\left(a\right)b+a\frac{d}{dx}\left(b\right)$
· Derivative of the linear function
$\frac{d}{dx}\left(x\right)=1$
· Derivative of the natural logarithm
$\frac{d}{dx}\left(\ln\left(x\right)\right)=\frac{1}{x}\frac{d}{dx}\left(x\right)$
· Derivative of a Constant
$\frac{d}{dx}\left(c\right)=0$
· Sum Rule for Differentiation
$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)$

Derivatives of trigonometric functions

· Derivative of the cosine function
$\frac{d}{dx}\left(\cos\left(\theta \right)\right)=-\frac{d}{dx}\left(\theta \right)\sin\left(\theta \right)$

Function Plot

Plotting: $\frac{1}{2}\left(-3\sin\left(3x\right)\ln\left(1+e^{5x}\right)+\frac{5e^{5x}\cos\left(3x\right)}{1+e^{5x}}\right)\left(1+e^{5x}\right)^{\frac{1}{2}\cos\left(3x\right)}$

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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

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Main Topic: Sum Rule of Differentiation

The sum rule is a method to find the derivative of a function that is the sum of two or more functions.

Used Formulas

See formulas (7)

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