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Step-by-step Solution

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

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Solution

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Final Answer

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

Problem to solve:

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

Choose the solving method

1

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\left(2x+1\right)^5$ and $g=\left(x^4-3\right)^6$

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

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

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

Subtract the values $5$ and $-1$

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

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

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

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

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

Subtract the values $5$ and $-1$

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

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

Subtract the values $6$ and $-1$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$10\left(2x+1\right)^{4}\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4-3\right)$
7

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

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

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

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

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

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

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

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

$10\left(2x+1\right)^{4}\left(x^4-3\right)^6+6\left(2x+1\right)^5\left(x^4-3\right)^{5}\frac{d}{dx}\left(x^4\right)$
8

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

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

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

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

Subtract the values $4$ and $-1$

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

Multiply $6$ times $4$

$10\left(2x+1\right)^{4}\left(x^4-3\right)^6+24x^{3}\left(2x+1\right)^5\left(x^4-3\right)^{5}$
9

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

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

Final Answer

$10\left(2x+1\right)^{4}\left(x^4-3\right)^6+24x^{3}\left(2x+1\right)^5\left(x^4-3\right)^{5}$
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θ
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<
>=
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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:

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

Main topic:

Product Rule of differentiation

Related Formulas:

6. See formulas

Time to solve it:

~ 0.11 s

Related topics:

Product Rule of differentiationDifferential CalculusBasic Differentiation RulesBasic DerivativesCalculusPower Rule for DerivativesSum Rule of DifferentiationConstant Rule

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