What is the derivative of (tan(x/2+cot(x/2))/x) ?

The derivative of (tan(x/2+cot(x/2))/x) is (xsec^2(x/2)-xcsc^2(x/2)-4csc(x))/(2x^2)

What is the first derivative of (tan(x/2+cot(x/2))/x) ?

The first derivative of (tan(x/2+cot(x/2))/x) is (xsec^2(x/2)-xcsc^2(x/2)-4csc(x))/(2x^2)

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\frac{d}{d x} (\frac{tan ( \frac{x}{2} ) + cot ( \frac{x}{2} )}{x})

\frac{x sec ^{2} ( \frac{x}{2} ) - x csc ^{2} ( \frac{x}{2} ) - 4 csc ( x )}{2 x ^{2}}

Solution steps

\frac{d}{d x} (\frac{tan ( \frac{x}{2} ) + cot ( \frac{x}{2} )}{x})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( tan ( \frac{x}{2} ) + cot ( \frac{x}{2} ) ) x - \frac{d x}{d x} ( tan ( \frac{x}{2} ) + cot ( \frac{x}{2} ) )}{x ^{2}}

\frac{d}{d x} (tan (\frac{x}{2}) + cot (\frac{x}{2})) = sec^{2} (\frac{x}{2}) \frac{1}{2} - csc^{2} (\frac{x}{2}) \frac{1}{2}

\frac{d x}{d x} = 1

= \frac{( sec ^{2} ( \frac{x}{2} ) \frac{1}{2} - csc ^{2} ( \frac{x}{2} ) \frac{1}{2} ) x - 1 \cdot ( tan ( \frac{x}{2} ) + cot ( \frac{x}{2} ) )}{x ^{2}}

Simplify

\frac{( sec ^{2} ( \frac{x}{2} ) \frac{1}{2} - csc ^{2} ( \frac{x}{2} ) \frac{1}{2} ) x - 1 \cdot ( tan ( \frac{x}{2} ) + cot ( \frac{x}{2} ) )}{x ^{2}} : \frac{x sec ^{2} ( \frac{x}{2} ) - x csc ^{2} ( \frac{x}{2} ) - 4 csc ( x )}{2 x ^{2}}

= \frac{x sec ^{2} ( \frac{x}{2} ) - x csc ^{2} ( \frac{x}{2} ) - 4 csc ( x )}{2 x ^{2}}

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

\frac{d}{d x} (1 - sin^{2} (x))

\int_{- 3}^{5} 6 - y - \frac{( y - 3 ) ^{2}}{4} d y

f (x) = 7 tan^{3} (x)

1 6^{x}

What is the derivative of (tan(x/2+cot(x/2))/x) ?
The derivative of (tan(x/2+cot(x/2))/x) is (xsec^2(x/2)-xcsc^2(x/2)-4csc(x))/(2x^2)
What is the first derivative of (tan(x/2+cot(x/2))/x) ?
The first derivative of (tan(x/2+cot(x/2))/x) is (xsec^2(x/2)-xcsc^2(x/2)-4csc(x))/(2x^2)