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

The derivative of x^{3/2}log_{2}(sqrt(x+1)) is (3x^{1/2}ln(x+1)(x+1)+2x^{3/2})/(4ln(2)(x+1))

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

The first derivative of x^{3/2}log_{2}(sqrt(x+1)) is (3x^{1/2}ln(x+1)(x+1)+2x^{3/2})/(4ln(2)(x+1))

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

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

Solution steps

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

Apply the Product Rule:

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

f = x^{\frac{3}{2}}, g = lo g_{2} (x + 1)

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

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

\frac{d}{d x} (lo g_{2} (x + 1)) = \frac{1}{2 ln ( 2 ) x + 1 ( x + 1 ) ^{\frac{1}{2}}}

= \frac{3 x ^{\frac{1}{2}}}{2} lo g_{2} (x + 1) + \frac{1}{2 ln ( 2 ) x + 1 ( x + 1 ) ^{\frac{1}{2}}} x^{\frac{3}{2}}

Simplify

\frac{3 x ^{\frac{1}{2}}}{2} lo g_{2} (x + 1) + \frac{1}{2 ln ( 2 ) x + 1 ( x + 1 ) ^{\frac{1}{2}}} x^{\frac{3}{2}} : \frac{3 x ^{\frac{1}{2}} ln ( x + 1 ) ( x + 1 ) + 2 x ^{\frac{3}{2}}}{4 ln ( 2 ) ( x + 1 )}

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

\int \frac{2}{3} x^{4} d x

\frac{\partial}{\partial x} (e^{y^{2}})

\int x (x + 1)^{100} d x

f (x) = e^{5 x}, at x = \frac{1}{5} ln (5)

x \to 4 - lim (\frac{1 - x}{4 - x})

What is the derivative of x^{3/2}log_{2}(sqrt(x+1)) ?
The derivative of x^{3/2}log_{2}(sqrt(x+1)) is (3x^{1/2}ln(x+1)(x+1)+2x^{3/2})/(4ln(2)(x+1))
What is the first derivative of x^{3/2}log_{2}(sqrt(x+1)) ?
The first derivative of x^{3/2}log_{2}(sqrt(x+1)) is (3x^{1/2}ln(x+1)(x+1)+2x^{3/2})/(4ln(2)(x+1))