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

Step-by-step Solution

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Final Answer

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

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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

Apply logarithm properties to both sides of the equality

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\frac{y^{\prime}}{y}=10\left(\frac{1}{2x+1}\right)\frac{d}{dx}\left(x\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4\right)$
16

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

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

Multiply the fraction and term

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

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

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

Multiply the fraction and term

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

Multiply both sides of the equation by $y$

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

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

Final Answer

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

Explore different ways to solve this problem

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 (2x+1)^5(x^4-3)^6 using the product ruleFind derivative of (2x+1)^5(x^4-3)^6 using the quotient ruleFind derivative of (2x+1)^5(x^4-3)^6 using the definition

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Function Plot

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

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0
a
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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

How to improve your answer:

Main Topic: Quotient of Powers

The quotient of powers of same base is equivalent to the same base to the power of the difference of exponents: $\frac{a^m}{a^n}=a^{m-n}$.

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