What is (\partial)/(\partial x)((0.09(5^x))/(x^3)) ?

The answer to (\partial)/(\partial x)((0.09(5^x))/(x^3)) is (0.09*5^x(1.60943…x-3))/(x^4)

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(\partial)/(\partial x)((0.09(5^x))/(x^3))

\frac{\partial}{\partial x} (\frac{0.09 ( 5 ^{x} )}{x ^{3}})

\frac{0.09 \cdot 5 ^{x} ( 1.60943 \dots x - 3 )}{x ^{4}}

Solution steps

\frac{\partial}{\partial x} (\frac{0.09 ( 5 ^{x} )}{x ^{3}})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 0.09 \frac{\partial}{\partial x} (\frac{5 ^{x}}{x ^{3}})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= 0.09 \cdot \frac{\frac{\partial}{\partial x} ( 5 ^{x} ) x ^{3} - \frac{\partial}{\partial x} ( x ^{3} ) 5 ^{x}}{( x ^{3} ) ^{2}}

\frac{\partial}{\partial x} (5^{x}) = 5^{x} ln (5)

\frac{\partial}{\partial x} (x^{3}) = 3 x^{2}

= 0.09 \cdot \frac{5 ^{x} ln ( 5 ) x ^{3} - 3 x ^{2} \cdot 5 ^{x}}{( x ^{3} ) ^{2}}

Simplify

0.09 \cdot \frac{5 ^{x} ln ( 5 ) x ^{3} - 3 x ^{2} \cdot 5 ^{x}}{( x ^{3} ) ^{2}} : \frac{0.09 \cdot 5 ^{x} ( 1.60943 \dots x - 3 )}{x ^{4}}

= \frac{0.09 \cdot 5 ^{x} ( 1.60943 \dots x - 3 )}{x ^{4}}

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What is (\partial)/(\partial x)((0.09(5^x))/(x^3)) ?
The answer to (\partial)/(\partial x)((0.09(5^x))/(x^3)) is (0.09*5^x(1.60943…x-3))/(x^4)