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
$\frac{d}{dx}\left(\left(2x+1\right)^5\left(x^4-3\right)^6\right)$
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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
$\ln\left(y\right)=\ln\left(\left(2x+1\right)^5\left(x^4-3\right)^6\right)$
Intermediate steps
Apply logarithm properties to both sides of the equality
$\ln\left(y\right)=\ln\left(\left(2x+1\right)^5\left(x^4-3\right)^6\right)$
Applying the product rule for logarithms: $\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)$
$\ln\left(y\right)=\ln\left(\left(2x+1\right)^5\right)+\ln\left(\left(x^4-3\right)^6\right)$
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
$\ln\left(y\right)=5\ln\left(2x+1\right)+\ln\left(\left(x^4-3\right)^6\right)$
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
$\ln\left(y\right)=5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)$
3
Apply logarithm properties to both sides of the equality
$\ln\left(y\right)=5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)$
Explain more
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 $\displaystyle f'(x)=\frac{a'}{a}$
$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)\right)$
Intermediate steps
The derivative of the linear function is equal to $1$
$1\left(\frac{1}{y}\right)$
Any expression multiplied by $1$ is equal to itself
$\frac{1}{y}$
6
The derivative of the linear function is equal to $1$
$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(5\ln\left(2x+1\right)+6\ln\left(x^4-3\right)\right)$
Explain more
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
$\frac{y^{\prime}}{y}=5\left(\frac{1}{2x+1}\right)\frac{d}{dx}\left(2x\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4\right)$
Intermediate steps
The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function
$10\left(\frac{1}{2x+1}\right)\frac{d}{dx}\left(x\right)$
The derivative of the linear function is equal to $1$
$10\left(\frac{1}{2x+1}\right)$
15
The derivative of the linear function times a constant, is equal to the constant
$\frac{y^{\prime}}{y}=10\left(\frac{1}{2x+1}\right)\frac{d}{dx}\left(x\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4\right)$
Explain more
Intermediate steps
The derivative of the linear function is equal to $1$
$1\left(\frac{1}{y}\right)$
Any expression multiplied by $1$ is equal to itself
$\frac{1}{y}$
The derivative of the linear function is equal to $1$
$10\cdot 1\left(\frac{1}{2x+1}\right)$
Any expression multiplied by $1$ is equal to itself
$10\left(\frac{1}{2x+1}\right)$
16
The derivative of the linear function is equal to $1$
$\frac{y^{\prime}}{y}=10\left(\frac{1}{2x+1}\right)+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4\right)$
Explain more
17
Multiply the fraction and term
$\frac{y^{\prime}}{y}=\frac{10}{2x+1}+6\left(\frac{1}{x^4-3}\right)\frac{d}{dx}\left(x^4\right)$
Intermediate steps
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$24\left(\frac{1}{x^4-3}\right)x^{\left(4-1\right)}$
Subtract the values $4$ and $-1$
$24\left(\frac{1}{x^4-3}\right)x^{3}$
18
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{y^{\prime}}{y}=\frac{10}{2x+1}+24\left(\frac{1}{x^4-3}\right)x^{3}$
Explain more
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$\left(\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}\right)\left(2x+1\right)^5\left(x^4-3\right)^6$