# Find the derivative of (3x^2)/(a^7^(1/5))(20-1e^(1/100x))

## \frac{d}{dx}\left(\frac{3x^2}{\sqrt[5]{a^{7}}}\cdot\left(20-e^{\frac{1}{100} x}\right)\right)

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$6xa^{-\frac{7}{5}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}3a^{-\frac{7}{5}}x^2e^{\frac{1}{100}x}$

## Step by step solution

Problem

$\frac{d}{dx}\left(\frac{3x^2}{\sqrt[5]{a^{7}}}\cdot\left(20-e^{\frac{1}{100} x}\right)\right)$
1

Applying the power of a power property

$\frac{d}{dx}\left(\frac{3x^2}{\sqrt[5]{a^{7}}}\left(20-e^{\frac{1}{100}x}\right)\right)$
2

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\frac{3x^2}{\sqrt[5]{a^{7}}}$ and $g=20-e^{\frac{1}{100}x}$

$\frac{3x^2}{\sqrt[5]{a^{7}}}\cdot\frac{d}{dx}\left(20-e^{\frac{1}{100}x}\right)+\left(20-e^{\frac{1}{100}x}\right)\frac{d}{dx}\left(\frac{3x^2}{\sqrt[5]{a^{7}}}\right)$
3

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{3x^2}{\sqrt[5]{a^{7}}}\cdot\frac{d}{dx}\left(20-e^{\frac{1}{100}x}\right)+\left(20-e^{\frac{1}{100}x}\right)\frac{\sqrt[5]{a^{7}}\cdot\frac{d}{dx}\left(3x^2\right)-3x^2\frac{d}{dx}\left(\sqrt[5]{a^{7}}\right)}{\left(\sqrt[5]{a^{7}}\right)^2}$
4

The derivative of the constant function is equal to zero

$\frac{3x^2}{\sqrt[5]{a^{7}}}\cdot\frac{d}{dx}\left(20-e^{\frac{1}{100}x}\right)+\left(20-e^{\frac{1}{100}x}\right)\frac{0\left(-3\right)x^2+\sqrt[5]{a^{7}}\cdot\frac{d}{dx}\left(3x^2\right)}{\left(\sqrt[5]{a^{7}}\right)^2}$
5

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

$\frac{3x^2}{\sqrt[5]{a^{7}}}\cdot\frac{d}{dx}\left(20-e^{\frac{1}{100}x}\right)+\left(20-e^{\frac{1}{100}x}\right)\frac{0+\sqrt[5]{a^{7}}\cdot\frac{d}{dx}\left(3x^2\right)}{\left(\sqrt[5]{a^{7}}\right)^2}$
6

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

$\frac{3x^2}{\sqrt[5]{a^{7}}}\cdot\frac{d}{dx}\left(20-e^{\frac{1}{100}x}\right)+\left(20-e^{\frac{1}{100}x}\right)\frac{0+3\sqrt[5]{a^{7}}\cdot\frac{d}{dx}\left(x^2\right)}{\left(\sqrt[5]{a^{7}}\right)^2}$
7

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

$\frac{3x^2}{\sqrt[5]{a^{7}}}\cdot\frac{d}{dx}\left(20-e^{\frac{1}{100}x}\right)+\frac{0+3\cdot 2x\sqrt[5]{a^{7}}}{\left(\sqrt[5]{a^{7}}\right)^2}\left(20-e^{\frac{1}{100}x}\right)$
8

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

$\frac{3x^2}{\sqrt[5]{a^{7}}}\left(\frac{d}{dx}\left(-e^{\frac{1}{100}x}\right)+\frac{d}{dx}\left(20\right)\right)+\frac{0+3\cdot 2x\sqrt[5]{a^{7}}}{\left(\sqrt[5]{a^{7}}\right)^2}\left(20-e^{\frac{1}{100}x}\right)$
9

The derivative of the constant function is equal to zero

$\frac{3x^2}{\sqrt[5]{a^{7}}}\left(\frac{d}{dx}\left(-e^{\frac{1}{100}x}\right)+0\right)+\frac{0+3\cdot 2x\sqrt[5]{a^{7}}}{\left(\sqrt[5]{a^{7}}\right)^2}\left(20-e^{\frac{1}{100}x}\right)$
10

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

$\frac{3x^2}{\sqrt[5]{a^{7}}}\left(0-\frac{d}{dx}\left(e^{\frac{1}{100}x}\right)\right)+\frac{0+3\cdot 2x\sqrt[5]{a^{7}}}{\left(\sqrt[5]{a^{7}}\right)^2}\left(20-e^{\frac{1}{100}x}\right)$
11

Applying the derivative of the exponential function

$\frac{3x^2}{\sqrt[5]{a^{7}}}\left(0-1\cdot 1\frac{d}{dx}\left(\frac{1}{100}x\right)e^{\frac{1}{100}x}\right)+\frac{0+3\cdot 2x\sqrt[5]{a^{7}}}{\left(\sqrt[5]{a^{7}}\right)^2}\left(20-e^{\frac{1}{100}x}\right)$
12

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

$\frac{3x^2}{\sqrt[5]{a^{7}}}\left(0-1\cdot 1\cdot \frac{1}{100}e^{\frac{1}{100}x}\cdot\frac{d}{dx}\left(x\right)\right)+\frac{0+3\cdot 2x\sqrt[5]{a^{7}}}{\left(\sqrt[5]{a^{7}}\right)^2}\left(20-e^{\frac{1}{100}x}\right)$
13

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

$\frac{3x^2}{\sqrt[5]{a^{7}}}\left(0-1\cdot 1\cdot 1\cdot \frac{1}{100}e^{\frac{1}{100}x}\right)+\frac{0+3\cdot 2x\sqrt[5]{a^{7}}}{\left(\sqrt[5]{a^{7}}\right)^2}\left(20-e^{\frac{1}{100}x}\right)$
14

Multiply $2$ times $3$

$\frac{3x^2}{\sqrt[5]{a^{7}}}\left(0-\frac{1}{100}e^{\frac{1}{100}x}\right)+\frac{0+6x\sqrt[5]{a^{7}}}{\left(\sqrt[5]{a^{7}}\right)^2}\left(20-e^{\frac{1}{100}x}\right)$
15

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

$\frac{6x\sqrt[5]{a^{7}}}{\left(\sqrt[5]{a^{7}}\right)^2}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}\cdot\frac{3x^2}{\sqrt[5]{a^{7}}}e^{\frac{1}{100}x}$
16

Applying the power of a power property

$\frac{6x\sqrt[5]{a^{7}}}{\sqrt[5]{a^{14}}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}\cdot\frac{3x^2}{\sqrt[5]{a^{7}}}e^{\frac{1}{100}x}$
17

Simplifying the fraction by $a$

$6xa^{\left(\frac{7}{5}-\frac{14}{5}\right)}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}\cdot\frac{3x^2}{\sqrt[5]{a^{7}}}e^{\frac{1}{100}x}$
18

Subtract the values $\frac{7}{5}$ and $-\frac{14}{5}$

$6xa^{-\frac{7}{5}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}\cdot\frac{3x^2}{\sqrt[5]{a^{7}}}e^{\frac{1}{100}x}$
19

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$6xa^{-\frac{7}{5}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}3a^{-\frac{7}{5}}x^2e^{\frac{1}{100}x}$
20

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$6x\frac{1}{\sqrt[5]{a^{7}}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}3\frac{1}{\sqrt[5]{a^{7}}}x^2e^{\frac{1}{100}x}$
21

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=6$ and $x=\sqrt[5]{a^{7}}$

$x\frac{6}{\sqrt[5]{a^{7}}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}x^2\frac{3}{\sqrt[5]{a^{7}}}e^{\frac{1}{100}x}$
22

Multiplying the fraction and term

$\frac{6x}{\sqrt[5]{a^{7}}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}\cdot\frac{3x^2}{\sqrt[5]{a^{7}}}e^{\frac{1}{100}x}$
23

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\frac{6x}{\sqrt[5]{a^{7}}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}3a^{-\frac{7}{5}}x^2e^{\frac{1}{100}x}$
24

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$6xa^{-\frac{7}{5}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}3a^{-\frac{7}{5}}x^2e^{\frac{1}{100}x}$

$6xa^{-\frac{7}{5}}\left(20-e^{\frac{1}{100}x}\right)-\frac{1}{100}3a^{-\frac{7}{5}}x^2e^{\frac{1}{100}x}$

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