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Find the derivative $\frac{d}{dx}\left(\sqrt[3]{x}\cos\left(x^2+e^{2x}\right)-3x\ln\left(\frac{3x}{x+1}\right)\right)$ using the sum rule

Step-by-step Solution

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Solution

Final answer to the problem

$\frac{\cos\left(x^2+e^{2x}\right)}{3\sqrt[3]{x^{2}}}+\sqrt[3]{x}\left(-2x-2e^{2x}\right)\sin\left(x^2+e^{2x}\right)-3\left(\ln\left(3x\right)+1\right)+3\left(\ln\left(x+1\right)+\frac{x}{x+1}\right)$
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Step-by-step Solution

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  • 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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1

Simplify the derivative by applying the properties of logarithms

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$\frac{d}{dx}\left(\sqrt[3]{x}\cos\left(x^2+e^{2x}\right)-3x\ln\left(3x\right)+3x\ln\left(x+1\right)\right)$

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Learn how to solve problems step by step online. Find the derivative d/dx(x^(1/3)cos(x^2+e^(2x))-3xln((3x)/(x+1))) using the sum rule. Simplify the derivative by applying the properties of logarithms. The derivative of a sum of two or more functions is the sum of the derivatives of each function. The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=\sqrt[3]{x} and g=\cos\left(x^2+e^{2x}\right).

Final answer to the problem

$\frac{\cos\left(x^2+e^{2x}\right)}{3\sqrt[3]{x^{2}}}+\sqrt[3]{x}\left(-2x-2e^{2x}\right)\sin\left(x^2+e^{2x}\right)-3\left(\ln\left(3x\right)+1\right)+3\left(\ln\left(x+1\right)+\frac{x}{x+1}\right)$

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Function Plot

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

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7
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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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