What is the derivative of (1+x^2(1+sqrt(x))) ?

The derivative of (1+x^2(1+sqrt(x))) is (5x^2+4xsqrt(x)+1)/(2sqrt(x))

What is the first derivative of (1+x^2(1+sqrt(x))) ?

The first derivative of (1+x^2(1+sqrt(x))) is (5x^2+4xsqrt(x)+1)/(2sqrt(x))

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

\frac{5 x ^{2} + 4 x x + 1}{2 x}

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

= \frac{5 x ^{2} + 4 x x + 1}{2 x}

What is the derivative of (1+x^2(1+sqrt(x))) ?
The derivative of (1+x^2(1+sqrt(x))) is (5x^2+4xsqrt(x)+1)/(2sqrt(x))
What is the first derivative of (1+x^2(1+sqrt(x))) ?
The first derivative of (1+x^2(1+sqrt(x))) is (5x^2+4xsqrt(x)+1)/(2sqrt(x))