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Find the derivative using logarithmic differentiation method $\frac{d}{dx}\left(x^{7x}\right)$

Step-by-step Solution

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Solution

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Final answer to the problem

$7\left(\ln\left(x\right)+1\right)x^{7x}$
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Step-by-step Solution

Specify the solving method

1

To derive the function $x^{7x}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=x^{7x}$
2

Apply natural logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(x^{7x}\right)$
3

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

$\ln\left(y\right)=7x\ln\left(x\right)$
4

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

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(7x\ln\left(x\right)\right)$
5

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=7\frac{d}{dx}\left(x\right)\ln\left(x\right)+x\left(7\frac{d}{dx}\left(\ln\left(x\right)\right)+\frac{d}{dx}\left(7\right)\ln\left(x\right)\right)$
6

The derivative of the constant function ($7$) is equal to zero

$\frac{d}{dx}\left(\ln\left(y\right)\right)=7\frac{d}{dx}\left(x\right)\ln\left(x\right)+x\left(7\frac{d}{dx}\left(\ln\left(x\right)\right)+0\ln\left(x\right)\right)$
7

Any expression multiplied by $0$ is equal to $0$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=7\frac{d}{dx}\left(x\right)\ln\left(x\right)+x\left(7\frac{d}{dx}\left(\ln\left(x\right)\right)+0\right)$
8

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

$\frac{d}{dx}\left(\ln\left(y\right)\right)=7\frac{d}{dx}\left(x\right)\ln\left(x\right)+7x\frac{d}{dx}\left(\ln\left(x\right)\right)$
9

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

$\frac{d}{dx}\left(\ln\left(y\right)\right)=7\ln\left(x\right)+7x\frac{d}{dx}\left(\ln\left(x\right)\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=7\ln\left(x\right)+7x\frac{1}{x}\frac{d}{dx}\left(x\right)$
11

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

$\frac{y^{\prime}}{y}=7\ln\left(x\right)+7x\frac{1}{x}\frac{d}{dx}\left(x\right)$
12

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

$\frac{y^{\prime}}{y}=7\ln\left(x\right)+7x\frac{1}{x}$
13

Multiply the fraction and term

$\frac{y^{\prime}}{y}=7\ln\left(x\right)+\frac{7x}{x}$
14

Simplify the fraction

$\frac{y^{\prime}}{y}=7\ln\left(x\right)+7$
15

Factor the polynomial $7\ln\left(x\right)+7$ by it's greatest common factor (GCF): $7$

$\frac{y^{\prime}}{y}=7\left(\ln\left(x\right)+1\right)$
16

Multiply both sides of the equation by $y$

$y^{\prime}=7\left(\ln\left(x\right)+1\right)y$
17

Substitute $y$ for the original function: $x^{7x}$

$y^{\prime}=7\left(\ln\left(x\right)+1\right)x^{7x}$
18

The derivative of the function results in

$7\left(\ln\left(x\right)+1\right)x^{7x}$

Final answer to the problem

$7\left(\ln\left(x\right)+1\right)x^{7x}$

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 x^7x using the quotient ruleFind derivative of x^7x using logarithmic differentiationFind derivative of x^7x using the definition

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Plotting: $7\left(\ln\left(x\right)+1\right)x^{7x}$

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◻√◻
◻√
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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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Main Topic: Logarithmic Differentiation

The logarithmic derivative of a function f(x) is defined by the formula f'(x)/f(x).

Used Formulas

4. See formulas

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