What is the derivative of (x^2+sqrt(x)+1/(x^{3/2)}) ?

The derivative of (x^2+sqrt(x)+1/(x^{3/2)}) is (x^{1/2}+x^2-3-3sqrt(x))/(2x^{5/2)}

What is the first derivative of (x^2+sqrt(x)+1/(x^{3/2)}) ?

The first derivative of (x^2+sqrt(x)+1/(x^{3/2)}) is (x^{1/2}+x^2-3-3sqrt(x))/(2x^{5/2)}

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\frac{d}{d x} (\frac{x ^{2} + x + 1}{x ^{\frac{3}{2}}})

\frac{x ^{\frac{1}{2}} + x ^{2} - 3 - 3 x}{2 x ^{\frac{5}{2}}}

Solution steps

\frac{d}{d x} (\frac{x ^{2} + x + 1}{x ^{\frac{3}{2}}})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( x ^{2} + x + 1 ) x ^{\frac{3}{2}} - \frac{d}{d x} ( x ^{\frac{3}{2}} ) ( x ^{2} + x + 1 )}{( x ^{\frac{3}{2}} ) ^{2}}

\frac{d}{d x} (x^{2} + x + 1) = \frac{1 + 4 x ^{\frac{3}{2}}}{2 x ^{\frac{1}{2}}}

\frac{d}{d x} (x^{\frac{3}{2}}) = \frac{3 x ^{\frac{1}{2}}}{2}

= \frac{\frac{1 + 4 x ^{\frac{3}{2}}}{2 x ^{\frac{1}{2}}} x ^{\frac{3}{2}} - \frac{3 x ^{\frac{1}{2}}}{2} ( x ^{2} + x + 1 )}{( x ^{\frac{3}{2}} ) ^{2}}

Simplify

\frac{\frac{1 + 4 x ^{\frac{3}{2}}}{2 x ^{\frac{1}{2}}} x ^{\frac{3}{2}} - \frac{3 x ^{\frac{1}{2}}}{2} ( x ^{2} + x + 1 )}{( x ^{\frac{3}{2}} ) ^{2}} : \frac{x ^{\frac{1}{2}} + x ^{2} - 3 - 3 x}{2 x ^{\frac{5}{2}}}

= \frac{x ^{\frac{1}{2}} + x ^{2} - 3 - 3 x}{2 x ^{\frac{5}{2}}}

(- 4, - 3), (3, - 3)

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

\int π^{2} - 1

\int \frac{( 3 x - 1 )}{x ^{2} - x - 6} d x

\int_{- \infty}^{0} z e^{2 z} d z

What is the derivative of (x^2+sqrt(x)+1/(x^{3/2)}) ?
The derivative of (x^2+sqrt(x)+1/(x^{3/2)}) is (x^{1/2}+x^2-3-3sqrt(x))/(2x^{5/2)}
What is the first derivative of (x^2+sqrt(x)+1/(x^{3/2)}) ?
The first derivative of (x^2+sqrt(x)+1/(x^{3/2)}) is (x^{1/2}+x^2-3-3sqrt(x))/(2x^{5/2)}