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

The derivative of (x+(2x)^{1/2})(2x+1) is (6x+4*2^{1/2}x^{3/2}+1)/(2^{1/2)x^{1/2}}+1

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

The first derivative of (x+(2x)^{1/2})(2x+1) is (6x+4*2^{1/2}x^{3/2}+1)/(2^{1/2)x^{1/2}}+1

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derivative of

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

\frac{6 x + 4 \cdot 2 ^{\frac{1}{2}} x ^{\frac{3}{2}} + 1}{2 ^{\frac{1}{2}} x ^{\frac{1}{2}}} + 1

Solution steps

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

Simplify

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

(1 + \frac{1}{2 ^{\frac{1}{2}} x ^{\frac{1}{2}}}) (2 x + 1) + 2 (x + 2^{\frac{1}{2}} x^{\frac{1}{2}}) : \frac{6 x + 4 \cdot 2 ^{\frac{1}{2}} x ^{\frac{3}{2}} + 1}{2 ^{\frac{1}{2}} x ^{\frac{1}{2}}} + 1

= \frac{6 x + 4 \cdot 2 ^{\frac{1}{2}} x ^{\frac{3}{2}} + 1}{2 ^{\frac{1}{2}} x ^{\frac{1}{2}}} + 1

\int_{0}^{\frac{π}{2}} sec^{2} (x) d x

\int \frac{x ^{3} + 27}{x + 3} d x

\int fdf

\int_{\frac{π}{4}}^{π} sin (x) d x

What is the derivative of (x+(2x)^{1/2})(2x+1) ?
The derivative of (x+(2x)^{1/2})(2x+1) is (6x+4*2^{1/2}x^{3/2}+1)/(2^{1/2)x^{1/2}}+1
What is the first derivative of (x+(2x)^{1/2})(2x+1) ?
The first derivative of (x+(2x)^{1/2})(2x+1) is (6x+4*2^{1/2}x^{3/2}+1)/(2^{1/2)x^{1/2}}+1