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

The derivative of x(x^2+1^{1/2}) is (2x^2+1)/((x^2+1)^{1/2)}

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

The first derivative of x(x^2+1^{1/2}) is (2x^2+1)/((x^2+1)^{1/2)}

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

= 1 \cdot (x^{2} + 1)^{\frac{1}{2}} + \frac{x}{( x ^{2} + 1 ) ^{\frac{1}{2}}} x

Simplify

1 \cdot (x^{2} + 1)^{\frac{1}{2}} + \frac{x}{( x ^{2} + 1 ) ^{\frac{1}{2}}} x : \frac{2 x ^{2} + 1}{( x ^{2} + 1 ) ^{\frac{1}{2}}}

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

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

n = 1 \sum \infty (1 - \frac{5}{n})^{n}

\frac{4}{3} π x^{3}

sin (2 t)

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