Step-by-step Solution

Find the derivative using logarithmic differentiation method $\frac{d}{dx}\left(\left(2x+1\right)^5\left(x^4-3\right)^6\right)$

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Final Answer

$\left(2x+1\right)^{5}\left(x^4-3\right)^{6}\left(\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}\right)$
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Step-by-step Solution

Problem to solve:

$\frac{d}{dx}\left(2x+1\right)^5\left(x^4-3\right)^6$

Solving method

1

To derive the function $\left(2x+1\right)^5\left(x^4-3\right)^6$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=\left(2x+1\right)^5\left(x^4-3\right)^6$
2

Apply natural logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(\left(2x+1\right)^5\left(x^4-3\right)^6\right)$
3

Applying the product rule for logarithms: $\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)$

$\ln\left(y\right)=\ln\left(\left(2x+1\right)^5\right)+\ln\left(\left(x^4-3\right)^6\right)$
4

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\ln\left(y\right)=5\ln\left(2x+1\right)+\ln\left(\left(x^4-3\right)^6\right)$
5

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\ln\left(y\right)=5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)$
6

Derive both sides of the equality with respect to $x$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)\right)$
7

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)\right)$

$1y^{\prime}\left(\frac{1}{y}\right)=\frac{d}{dx}\left(5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)\right)$

Any expression multiplied by $1$ is equal to itself

$y^{\prime}\frac{1}{y}=\frac{d}{dx}\left(5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)\right)$
8

The derivative of the linear function is equal to $1$

$y^{\prime}\frac{1}{y}=\frac{d}{dx}\left(5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)\right)$
9

The derivative of a sum of two functions is the sum of the derivatives of each function

$y^{\prime}\frac{1}{y}=\frac{d}{dx}\left(5\ln\left(2x+1\right)\right)+\frac{d}{dx}\left(6\ln\left(x^4-3\right)\right)$
10

The derivative of a function multiplied by a constant ($5$) is equal to the constant times the derivative of the function

$y^{\prime}\frac{1}{y}=5\frac{d}{dx}\left(\ln\left(2x+1\right)\right)+\frac{d}{dx}\left(6\ln\left(x^4-3\right)\right)$
11

The derivative of a function multiplied by a constant ($6$) is equal to the constant times the derivative of the function

$y^{\prime}\frac{1}{y}=5\frac{d}{dx}\left(\ln\left(2x+1\right)\right)+6\frac{d}{dx}\left(\ln\left(x^4-3\right)\right)$
12

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 $\displaystyle f'(x)=\frac{a'}{a}$

$y^{\prime}\frac{1}{y}=5\left(\frac{1}{2x+1}\right)\frac{d}{dx}\left(2x+1\right)+6\frac{d}{dx}\left(\ln\left(x^4-3\right)\right)$
13

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 $\displaystyle f'(x)=\frac{a'}{a}$

$y^{\prime}\frac{1}{y}=5\left(\frac{1}{2x+1}\right)\frac{d}{dx}\left(2x+1\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$
14

The derivative of a sum of two functions is the sum of the derivatives of each function

$y^{\prime}\frac{1}{y}=5\left(\frac{1}{2x+1}\right)\left(\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(1\right)\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$

$y^{\prime}\frac{1}{y}=5\left(\frac{1}{2x+1}\right)\left(\frac{d}{dx}\left(2x\right)+0\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$

$x+0=x$, where $x$ is any expression

$y^{\prime}\frac{1}{y}=5\left(\frac{1}{2x+1}\right)\frac{d}{dx}\left(2x\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$
15

The derivative of the constant function ($1$) is equal to zero

$y^{\prime}\frac{1}{y}=5\left(\frac{1}{2x+1}\right)\frac{d}{dx}\left(2x\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$

$y^{\prime}\frac{1}{y}=5\cdot 2\left(\frac{1}{2x+1}\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$

Multiply $5$ times $2$

$y^{\prime}\frac{1}{y}=10\left(\frac{1}{2x+1}\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$

Multiply the fraction and term

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

$\frac{10}{2x+1}\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$\frac{10}{2x+1}$
16

The derivative of the linear function times a constant, is equal to the constant

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$
17

The derivative of a sum of two functions is the sum of the derivatives of each function

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+6\left(\frac{1}{x^4-3}\right)\left(\frac{d}{dx}\left(x^4\right)+\frac{d}{dx}\left(-3\right)\right)$

$y^{\prime}\frac{1}{y}=5\left(\frac{1}{2x+1}\right)\left(\frac{d}{dx}\left(2x\right)+0\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$

$x+0=x$, where $x$ is any expression

$y^{\prime}\frac{1}{y}=5\left(\frac{1}{2x+1}\right)\frac{d}{dx}\left(2x\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4-3\right)$

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+6\left(\frac{1}{x^4-3}\right)\left(\frac{d}{dx}\left(x^4\right)+0\right)$

$x+0=x$, where $x$ is any expression

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4\right)$
18

The derivative of the constant function ($-3$) is equal to zero

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4\right)$

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+6\cdot 4x^{\left(4-1\right)}\left(\frac{1}{x^4-3}\right)$

Subtract the values $4$ and $-1$

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+6\cdot 4x^{3}\left(\frac{1}{x^4-3}\right)$

Multiply $6$ times $4$

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+24x^{3}\left(\frac{1}{x^4-3}\right)$

Multiply the fraction and term

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}$
19

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$y^{\prime}\frac{1}{y}=\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}$

$y^{\prime}=\frac{\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}}{\frac{1}{y}}$

Divide fractions $\frac{\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}}{\frac{1}{y}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

$y^{\prime}=y\left(\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}\right)$
20

Isolate $y'$

$y^{\prime}=y\left(\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}\right)$
21

Substitute $y$ for the original function: $\left(2x+1\right)^5\left(x^4-3\right)^6$

$y^{\prime}=\left(\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}\right)\left(2x+1\right)^5\left(x^4-3\right)^6$
22

The derivative of the function results in

$\left(\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}\right)\left(2x+1\right)^5\left(x^4-3\right)^6$

Multiplying polynomials $\left(2x+1\right)^5\left(x^4-3\right)^6$ and $\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}$

$\left(2x+1\right)^5\left(x^4-3\right)^6\frac{10}{2x+1}+\left(2x+1\right)^5\left(x^4-3\right)^6\frac{24x^{3}}{x^4-3}$

Factor the polynomial $\left(2x+1\right)^5\left(x^4-3\right)^6\frac{10}{2x+1}+\left(2x+1\right)^5\left(x^4-3\right)^6\frac{24x^{3}}{x^4-3}$ by it's GCF: $\left(2x+1\right)^{5}\left(x^4-3\right)^{6}$

$\left(2x+1\right)^{5}\left(x^4-3\right)^{6}\left(\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}\right)$
23

Simplifying

$\left(2x+1\right)^{5}\left(x^4-3\right)^{6}\left(\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}\right)$

Final Answer

$\left(2x+1\right)^{5}\left(x^4-3\right)^{6}\left(\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}\right)$
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0
a
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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

Tips on how to improve your answer:

$\frac{d}{dx}\left(2x+1\right)^5\left(x^4-3\right)^6$

Related Formulas:

6. See formulas

Time to solve it:

~ 0.22 s