What is the derivative of ((cos(x)/(1+sin(x)))^5) ?

The derivative of ((cos(x)/(1+sin(x)))^5) is -(5cos^4(x))/((1+sin(x))^5)

What is the first derivative of ((cos(x)/(1+sin(x)))^5) ?

The first derivative of ((cos(x)/(1+sin(x)))^5) is -(5cos^4(x))/((1+sin(x))^5)

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\frac{d}{d x} ((\frac{cos ( x )}{1 + sin ( x )})^{5})

- \frac{5 cos ^{4} ( x )}{( 1 + sin ( x ) ) ^{5}}

Solution steps

\frac{d}{d x} ((\frac{cos ( x )}{1 + sin ( x )})^{5})

Apply the chain rule:

5 (\frac{cos ( x )}{1 + sin ( x )})^{4} \frac{d}{d x} (\frac{cos ( x )}{1 + sin ( x )})

= 5 (\frac{cos ( x )}{1 + sin ( x )})^{4} \frac{d}{d x} (\frac{cos ( x )}{1 + sin ( x )})

\frac{d}{d x} (\frac{cos ( x )}{1 + sin ( x )}) = - \frac{1}{1 + sin ( x )}

= 5 (\frac{cos ( x )}{1 + sin ( x )})^{4} (- \frac{1}{1 + sin ( x )})

Simplify

5 (\frac{cos ( x )}{1 + sin ( x )})^{4} (- \frac{1}{1 + sin ( x )}) : - \frac{5 cos ^{4} ( x )}{( 1 + sin ( x ) ) ^{5}}

= - \frac{5 cos ^{4} ( x )}{( 1 + sin ( x ) ) ^{5}}

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What is the derivative of ((cos(x)/(1+sin(x)))^5) ?
The derivative of ((cos(x)/(1+sin(x)))^5) is -(5cos^4(x))/((1+sin(x))^5)
What is the first derivative of ((cos(x)/(1+sin(x)))^5) ?
The first derivative of ((cos(x)/(1+sin(x)))^5) is -(5cos^4(x))/((1+sin(x))^5)