What is the derivative of (sin(x/4+cos(x))^{1/x}) ?

The derivative of (sin(x/4+cos(x))^{1/x}) is

What is the first derivative of (sin(x/4+cos(x))^{1/x}) ?

The first derivative of (sin(x/4+cos(x))^{1/x}) is

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

(- \frac{ln ( sin ( \frac{x}{4} ) + cos ( x ) )}{x ^{2}} + \frac{\frac{1}{4} cos ( \frac{x}{4} ) - sin ( x )}{x ( sin ( \frac{x}{4} ) + cos ( x ) )}) (sin (\frac{x}{4}) + cos (x))^{\frac{1}{x}}

Solution steps

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

Apply exponent rule:

a^{b} = e^{b l n (a)}

(sin (\frac{x}{4}) + cos (x))^{\frac{1}{x}} = e^{\frac{1}{x} l n (s i n (\frac{x}{4}) + c o s (x))}

= \frac{d}{d x} (e^{\frac{1}{x} l n (s i n (\frac{x}{4}) + c o s (x))})

Apply the chain rule:

e^{\frac{1}{x} l n (s i n (\frac{x}{4}) + c o s (x))} \frac{d}{d x} (\frac{1}{x} ln (sin (\frac{x}{4}) + cos (x)))

= e^{\frac{1}{x} l n (s i n (\frac{x}{4}) + c o s (x))} \frac{d}{d x} (\frac{1}{x} ln (sin (\frac{x}{4}) + cos (x)))

\frac{d}{d x} (\frac{1}{x} ln (sin (\frac{x}{4}) + cos (x))) = - \frac{ln ( sin ( \frac{x}{4} ) + cos ( x ) )}{x ^{2}} + \frac{\frac{1}{4} cos ( \frac{x}{4} ) - sin ( x )}{x ( sin ( \frac{x}{4} ) + cos ( x ) )}

= e^{\frac{1}{x} l n (s i n (\frac{x}{4}) + c o s (x))} (- \frac{ln ( sin ( \frac{x}{4} ) + cos ( x ) )}{x ^{2}} + \frac{\frac{1}{4} cos ( \frac{x}{4} ) - sin ( x )}{x ( sin ( \frac{x}{4} ) + cos ( x ) )})

e^{\frac{1}{x} l n (s i n (\frac{x}{4}) + c o s (x))} = (sin (\frac{x}{4}) + cos (x))^{\frac{1}{x}}

= (- \frac{ln ( sin ( \frac{x}{4} ) + cos ( x ) )}{x ^{2}} + \frac{\frac{1}{4} cos ( \frac{x}{4} ) - sin ( x )}{x ( sin ( \frac{x}{4} ) + cos ( x ) )}) (sin (\frac{x}{4}) + cos (x))^{\frac{1}{x}}

(1 x + y e^{\frac{y}{x}}) d x - x e^{\frac{y}{x}} d y = 0

x \to 8 + lim (\frac{3}{x - 8})

\int ln ((7 x)^{2}) d x

\frac{\partial}{\partial x} (cos (2 x + 4 y))

\frac{d}{d x} (- \frac{x ^{2}}{2})

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