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Find the derivative of $\ln\left(x\right)\left(\sec\left(2x\right)+\tan\left(2x\right)\right)$

Step-by-step Solution

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Solving: $\frac{d}{dx}\left(\ln\left(x\right)\left(\sec\left(2x\right)+\tan\left(2x\right)\right)\right)$

Final Answer

$\frac{\sec\left(2x\right)+\tan\left(2x\right)}{x}+\ln\left(x\right)\left(2\sec\left(2x\right)\tan\left(2x\right)+2\sec\left(2x\right)^2\right)$
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Step-by-step Solution

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Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\ln\left(x\right)$ and $g=\sec\left(2x\right)+\tan\left(2x\right)$

$\frac{d}{dx}\left(\ln\left(x\right)\right)\left(\sec\left(2x\right)+\tan\left(2x\right)\right)+\ln\left(x\right)\frac{d}{dx}\left(\sec\left(2x\right)+\tan\left(2x\right)\right)$

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$\frac{d}{dx}\left(\ln\left(x\right)\right)\left(\sec\left(2x\right)+\tan\left(2x\right)\right)+\ln\left(x\right)\frac{d}{dx}\left(\sec\left(2x\right)+\tan\left(2x\right)\right)$

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Learn how to solve differential calculus problems step by step online. Find the derivative of ln(x)(sec(2x)+tan(2x)). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=\ln\left(x\right) and g=\sec\left(2x\right)+\tan\left(2x\right). 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}. Multiply the fraction and term. The derivative of a sum of two or more functions is the sum of the derivatives of each function.

Final Answer

$\frac{\sec\left(2x\right)+\tan\left(2x\right)}{x}+\ln\left(x\right)\left(2\sec\left(2x\right)\tan\left(2x\right)+2\sec\left(2x\right)^2\right)$

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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 lnx(sec2x+tan2x) using the product ruleFind derivative of lnx(sec2x+tan2x) using the quotient ruleFind derivative of lnx(sec2x+tan2x) using logarithmic differentiation

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Function Plot

Plotting: $\frac{\sec\left(2x\right)+\tan\left(2x\right)}{x}+\ln\left(x\right)\left(2\sec\left(2x\right)\tan\left(2x\right)+2\sec\left(2x\right)^2\right)$

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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
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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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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