What is the derivative of (e^{3x}sin(2x)/(5^x)) ?

The derivative of (e^{3x}sin(2x)/(5^x)) is (e^{3x}(2cos(2x)+3sin(2x)-ln(5)sin(2x)))/(5^x)

What is the first derivative of (e^{3x}sin(2x)/(5^x)) ?

The first derivative of (e^{3x}sin(2x)/(5^x)) is (e^{3x}(2cos(2x)+3sin(2x)-ln(5)sin(2x)))/(5^x)

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\frac{d}{d x} (\frac{e ^{3 x} sin ( 2 x )}{5 ^{x}})

\frac{e ^{3 x} ( 2 cos ( 2 x ) + 3 sin ( 2 x ) - ln ( 5 ) sin ( 2 x ) )}{5 ^{x}}

Solution steps

\frac{d}{d x} (\frac{e ^{3 x} sin ( 2 x )}{5 ^{x}})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( e ^{3 x} sin ( 2 x ) ) 5 ^{x} - \frac{d}{d x} ( 5 ^{x} ) e ^{3 x} sin ( 2 x )}{( 5 ^{x} ) ^{2}}

\frac{d}{d x} (e^{3 x} sin (2 x)) = e^{3 x} \cdot 3 sin (2 x) + cos (2 x) \cdot 2 e^{3 x}

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

= \frac{( e ^{3 x} \cdot 3 sin ( 2 x ) + cos ( 2 x ) \cdot 2 e ^{3 x} ) \cdot 5 ^{x} - 5 ^{x} ln ( 5 ) e ^{3 x} sin ( 2 x )}{( 5 ^{x} ) ^{2}}

Simplify

\frac{( e ^{3 x} \cdot 3 sin ( 2 x ) + cos ( 2 x ) \cdot 2 e ^{3 x} ) \cdot 5 ^{x} - 5 ^{x} ln ( 5 ) e ^{3 x} sin ( 2 x )}{( 5 ^{x} ) ^{2}} : \frac{e ^{3 x} ( 2 cos ( 2 x ) + 3 sin ( 2 x ) - ln ( 5 ) sin ( 2 x ) )}{5 ^{x}}

= \frac{e ^{3 x} ( 2 cos ( 2 x ) + 3 sin ( 2 x ) - ln ( 5 ) sin ( 2 x ) )}{5 ^{x}}

What is the derivative of (e^{3x}sin(2x)/(5^x)) ?
The derivative of (e^{3x}sin(2x)/(5^x)) is (e^{3x}(2cos(2x)+3sin(2x)-ln(5)sin(2x)))/(5^x)
What is the first derivative of (e^{3x}sin(2x)/(5^x)) ?
The first derivative of (e^{3x}sin(2x)/(5^x)) is (e^{3x}(2cos(2x)+3sin(2x)-ln(5)sin(2x)))/(5^x)