What is the derivative of f(x)=(sin(4x))/(sin(5x)) ?

The derivative of f(x)=(sin(4x))/(sin(5x)) is 4cos(4x)csc(5x)-5cot(5x)csc(5x)sin(4x)

What is the first derivative of f(x)=(sin(4x))/(sin(5x)) ?

The first derivative of f(x)=(sin(4x))/(sin(5x)) is 4cos(4x)csc(5x)-5cot(5x)csc(5x)sin(4x)

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derivative of

f (x) = \frac{sin ( 4 x )}{sin ( 5 x )}

4 cos (4 x) csc (5 x) - 5 cot (5 x) csc (5 x) sin (4 x)

Solution steps

\frac{d}{d x} (\frac{sin ( 4 x )}{sin ( 5 x )})

Simplify

\frac{sin ( 4 x )}{sin ( 5 x )} : sin (4 x) csc (5 x)

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

Apply the Product Rule:

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

f = sin (4 x), g = csc (5 x)

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

\frac{d}{d x} (sin (4 x)) = cos (4 x) \cdot 4

\frac{d}{d x} (csc (5 x)) = - cot (5 x) csc (5 x) \cdot 5

= cos (4 x) \cdot 4 csc (5 x) + (- cot (5 x) csc (5 x) \cdot 5) sin (4 x)

Simplify

= 4 cos (4 x) csc (5 x) - 5 cot (5 x) csc (5 x) sin (4 x)

y = arctan (3 x^{2})

\int e^{10 t} d t

\frac{d}{d x} (f (x)^{- 1} (x))

x^{2} y^{^{'}} + 2 x y = 21 cos^{2} (x)

\frac{x}{x - 1}

What is the derivative of f(x)=(sin(4x))/(sin(5x)) ?
The derivative of f(x)=(sin(4x))/(sin(5x)) is 4cos(4x)csc(5x)-5cot(5x)csc(5x)sin(4x)
What is the first derivative of f(x)=(sin(4x))/(sin(5x)) ?
The first derivative of f(x)=(sin(4x))/(sin(5x)) is 4cos(4x)csc(5x)-5cot(5x)csc(5x)sin(4x)