What is the derivative of (4csc(x)/(5^x)) ?

The derivative of (4csc(x)/(5^x)) is -(4csc(x)(cot(x)+ln(5)))/(5^x)

What is the first derivative of (4csc(x)/(5^x)) ?

The first derivative of (4csc(x)/(5^x)) is -(4csc(x)(cot(x)+ln(5)))/(5^x)

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\frac{d}{d x} (\frac{4 csc ( x )}{5 ^{x}})

- \frac{4 csc ( x ) ( cot ( x ) + ln ( 5 ) )}{5 ^{x}}

Solution steps

\frac{d}{d x} (\frac{4 csc ( x )}{5 ^{x}})

Take the constant out:

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

= 4 \frac{d}{d x} (\frac{csc ( x )}{5 ^{x}})

Apply the Quotient Rule:

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

= 4 \cdot \frac{\frac{d}{d x} ( csc ( x ) ) 5 ^{x} - \frac{d}{d x} ( 5 ^{x} ) csc ( x )}{( 5 ^{x} ) ^{2}}

\frac{d}{d x} (csc (x)) = - cot (x) csc (x)

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

= 4 \cdot \frac{( - cot ( x ) csc ( x ) ) \cdot 5 ^{x} - 5 ^{x} ln ( 5 ) csc ( x )}{( 5 ^{x} ) ^{2}}

Simplify

4 \cdot \frac{( - cot ( x ) csc ( x ) ) \cdot 5 ^{x} - 5 ^{x} ln ( 5 ) csc ( x )}{( 5 ^{x} ) ^{2}} : - \frac{4 csc ( x ) ( cot ( x ) + ln ( 5 ) )}{5 ^{x}}

= - \frac{4 csc ( x ) ( cot ( x ) + ln ( 5 ) )}{5 ^{x}}

What is the derivative of (4csc(x)/(5^x)) ?
The derivative of (4csc(x)/(5^x)) is -(4csc(x)(cot(x)+ln(5)))/(5^x)
What is the first derivative of (4csc(x)/(5^x)) ?
The first derivative of (4csc(x)/(5^x)) is -(4csc(x)(cot(x)+ln(5)))/(5^x)