What is the derivative of (sin(x)/(cos^3(x))) ?

The derivative of (sin(x)/(cos^3(x))) is sec^2(x)(1+3tan^2(x))

What is the first derivative of (sin(x)/(cos^3(x))) ?

The first derivative of (sin(x)/(cos^3(x))) is sec^2(x)(1+3tan^2(x))

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\frac{d}{d x} (\frac{sin ( x )}{cos ^{3} ( x )})

sec^{2} (x) (1 + 3 tan^{2} (x))

Solution steps

\frac{d}{d x} (\frac{sin ( x )}{cos ^{3} ( x )})

Simplify

\frac{sin ( x )}{cos ^{3} ( x )} : sec^{3} (x) sin (x)

= \frac{d}{d x} (sec^{3} (x) sin (x))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = sec^{3} (x), g = sin (x)

= \frac{d}{d x} (sec^{3} (x)) sin (x) + \frac{d}{d x} (sin (x)) sec^{3} (x)

\frac{d}{d x} (sec^{3} (x)) = 3 sec^{3} (x) tan (x)

\frac{d}{d x} (sin (x)) = cos (x)

= 3 sec^{3} (x) tan (x) sin (x) + cos (x) sec^{3} (x)

Simplify

3 sec^{3} (x) tan (x) sin (x) + cos (x) sec^{3} (x) : sec^{2} (x) (1 + 3 tan^{2} (x))

= sec^{2} (x) (1 + 3 tan^{2} (x))

What is the derivative of (sin(x)/(cos^3(x))) ?
The derivative of (sin(x)/(cos^3(x))) is sec^2(x)(1+3tan^2(x))
What is the first derivative of (sin(x)/(cos^3(x))) ?
The first derivative of (sin(x)/(cos^3(x))) is sec^2(x)(1+3tan^2(x))