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

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Solution

Final answer to the problem

$\left(5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x^2-2x\right)\left(x^2+3\right)^{\left(5x-2\right)}$
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Step-by-step Solution

How should I solve this problem?

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  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative using logarithmic differentiation
  • Find the derivative
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
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1

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

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

Apply natural logarithm to both sides of the equality

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$5\frac{d}{dx}\left(x\right)\ln\left(x^2+3\right)$

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

$5\ln\left(x^2+3\right)$
10

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

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

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

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

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

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

Subtract the values $2$ and $-1$

$2\left(5x-1\right)\frac{1}{x^2+3}x$
12

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

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

Multiply the fraction by the term

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

Any expression multiplied by $1$ is equal to itself

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

Multiply the fraction by the term

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

Multiply both sides of the equation by $y$

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

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

$y^{\prime}=\left(5\ln\left(x^2+3\right)+\frac{2\left(5x-1\right)x}{x^2+3}\right)\left(x^2+3\right)^{\left(5x-1\right)}$
16

The derivative of the function results in

$\left(5\ln\left(x^2+3\right)+\frac{2\left(5x-1\right)x}{x^2+3}\right)\left(x^2+3\right)^{\left(5x-1\right)}$

Combine all terms into a single fraction with $x^2+3$ as common denominator

$\frac{5\left(x^2+3\right)\ln\left(x^2+3\right)+2\left(5x-1\right)x}{x^2+3}\left(x^2+3\right)^{\left(5x-1\right)}$

Solve the product $5\left(x^2+3\right)\ln\left(x^2+3\right)$

$\frac{5x^2\ln\left(x^2+3\right)+5\cdot 3\ln\left(x^2+3\right)+2\left(5x-1\right)x}{x^2+3}\left(x^2+3\right)^{\left(5x-1\right)}$

Multiply $5$ times $3$

$\frac{5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+2\left(5x-1\right)x}{x^2+3}\left(x^2+3\right)^{\left(5x-1\right)}$

Solve the product $2\left(5x-1\right)x$

$\frac{5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+2\cdot 5x\cdot x+2\cdot -1x}{x^2+3}\left(x^2+3\right)^{\left(5x-1\right)}$

Solve the product $5\left(x^2+3\right)\ln\left(x^2+3\right)$

$\frac{5x^2\ln\left(x^2+3\right)+5\cdot 3\ln\left(x^2+3\right)+2\left(5x-1\right)x}{x^2+3}\left(x^2+3\right)^{\left(5x-1\right)}$

Multiply $5$ times $3$

$\frac{5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+2\left(5x-1\right)x}{x^2+3}\left(x^2+3\right)^{\left(5x-1\right)}$

Multiply $2$ times $5$

$\frac{5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x\cdot x+2\cdot -1x}{x^2+3}\left(x^2+3\right)^{\left(5x-1\right)}$

Multiply $2$ times $-1$

$\frac{5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x\cdot x-2x}{x^2+3}\left(x^2+3\right)^{\left(5x-1\right)}$

When multiplying two powers that have the same base ($x$), you can add the exponents

$\frac{5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x^2-2x}{x^2+3}\left(x^2+3\right)^{\left(5x-1\right)}$

Solve the product $2\left(5x-1\right)x$

$2\cdot 5x\cdot x+2\cdot -1x$

When multiplying two powers that have the same base ($x$), you can add the exponents

$2\cdot 5x^2+2\cdot -1x$

Multiplying the fraction by $\left(x^2+3\right)^{\left(5x-1\right)}$

$\frac{\left(5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x^2-2x\right)\left(x^2+3\right)^{\left(5x-1\right)}}{x^2+3}$

Simplify the fraction $\frac{\left(5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x^2-2x\right)\left(x^2+3\right)^{\left(5x-1\right)}}{x^2+3}$ by $x^2+3$

$\left(5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x^2-2x\right)\left(x^2+3\right)^{\left(5x-2\right)}$
17

Simplify the derivative

$\left(5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x^2-2x\right)\left(x^2+3\right)^{\left(5x-2\right)}$

Final answer to the problem

$\left(5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x^2-2x\right)\left(x^2+3\right)^{\left(5x-2\right)}$

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Function Plot

Plotting: $\left(5x^2\ln\left(x^2+3\right)+15\ln\left(x^2+3\right)+10x^2-2x\right)\left(x^2+3\right)^{\left(5x-2\right)}$

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a
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m
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x
y
z
.
(◻)
+
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×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
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

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