What is the derivative of ((1-4e^x)/(1-5cos(x))) ?

The derivative of ((1-4e^x)/(1-5cos(x))) is (-4e^x(1-5cos(x))-5sin(x)(1-4e^x))/((1-5cos(x))^2)

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

The first derivative of ((1-4e^x)/(1-5cos(x))) is (-4e^x(1-5cos(x))-5sin(x)(1-4e^x))/((1-5cos(x))^2)

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\frac{d}{d x} (\frac{( 1 - 4 e ^{x} )}{1 - 5 cos ( x )})

\frac{- 4 e ^{x} ( 1 - 5 cos ( x ) ) - 5 sin ( x ) ( 1 - 4 e ^{x} )}{( 1 - 5 cos ( x ) ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{( 1 - 4 e ^{x} )}{1 - 5 cos ( x )})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( 1 - 4 e ^{x} ) ( 1 - 5 cos ( x ) ) - \frac{d}{d x} ( 1 - 5 cos ( x ) ) ( 1 - 4 e ^{x} )}{( 1 - 5 cos ( x ) ) ^{2}}

\frac{d}{d x} (1 - 4 e^{x}) = - 4 e^{x}

\frac{d}{d x} (1 - 5 cos (x)) = 5 sin (x)

= \frac{( - 4 e ^{x} ) ( 1 - 5 cos ( x ) ) - 5 sin ( x ) ( 1 - 4 e ^{x} )}{( 1 - 5 cos ( x ) ) ^{2}}

Simplify

= \frac{- 4 e ^{x} ( 1 - 5 cos ( x ) ) - 5 sin ( x ) ( 1 - 4 e ^{x} )}{( 1 - 5 cos ( x ) ) ^{2}}

x \to - 8 lim (10 x + 1)

(x^{2} + 1) \cdot \frac{d y}{d x} + 6 x (y - 1) = 0

10 y^{^{''}} - 10 y^{^{'}} + 25 y = 0

f (x) = (\frac{2 + 3 x}{3 x}) (9 - 4 x)

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

What is the derivative of ((1-4e^x)/(1-5cos(x))) ?
The derivative of ((1-4e^x)/(1-5cos(x))) is (-4e^x(1-5cos(x))-5sin(x)(1-4e^x))/((1-5cos(x))^2)
What is the first derivative of ((1-4e^x)/(1-5cos(x))) ?
The first derivative of ((1-4e^x)/(1-5cos(x))) is (-4e^x(1-5cos(x))-5sin(x)(1-4e^x))/((1-5cos(x))^2)