Final Answer
Step-by-step Solution
Problem to solve:
Choose the solving method
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
Apply natural logarithm to both sides of the equality
Applying the product rule for logarithms: $\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)$
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
Derive both sides of the equality with respect to $x$
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 $\displaystyle f'(x)=\frac{a'}{a}$
The derivative of the linear function is equal to $1$
Any expression multiplied by $1$ is equal to itself
The derivative of the linear function is equal to $1$
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 ($5$) is equal to the constant times the derivative of the function
The derivative of a function multiplied by a constant ($6$) is equal to the constant times the derivative of the function
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 $\displaystyle f'(x)=\frac{a'}{a}$
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 $\displaystyle f'(x)=\frac{a'}{a}$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of the constant function ($1$) is equal to zero
$x+0=x$, where $x$ is any expression
The derivative of the constant function ($1$) is equal to zero
The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function
The derivative of the linear function is equal to $1$
The derivative of the linear function times a constant, is equal to the constant
Multiplying the fraction by $10$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of the constant function ($1$) is equal to zero
$x+0=x$, where $x$ is any expression
$x+0=x$, where $x$ is any expression
The derivative of the constant function ($-3$) is equal to zero
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
Subtract the values $4$ and $-1$
Multiply $6$ times $4$
Multiplying the fraction by $24$
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
Isolate $y'$
Divide fractions $\frac{\frac{10}{2x+1}+\frac{24x^{3}}{x^4-3}}{\frac{1}{y}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Isolate $y'$
Substitute $y$ for the original function: $\left(2x+1\right)^5\left(x^4-3\right)^6$
The derivative of the function results in
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
Obtained the least common multiple, we place the LCM as the denominator of each fraction and in the numerator of each fraction we add the factors that we need to complete
Simplify the numerators
Combine and simplify all terms in the same fraction with common denominator $\left(2x+1\right)\left(x^4-3\right)$
Multiplying the fraction by $\left(2x+1\right)^5$
Multiplying the fraction by $\left(x^4-3\right)^6$
Simplify the fraction $\frac{\left(58x^{4}-30+24x^{3}\right)\left(2x+1\right)^5\left(x^4-3\right)^6}{\left(2x+1\right)\left(x^4-3\right)}$ by $2x+1$
Simplify the fraction $\frac{\left(58x^{4}-30+24x^{3}\right)\left(2x+1\right)^{4}\left(x^4-3\right)^6}{x^4-3}$ by $x^4-3$
Simplifying