What is the derivative of sqrt(x)(x^3-5x^2+1) ?

The derivative of sqrt(x)(x^3-5x^2+1) is (7x^3-25x^2+1)/(2sqrt(x))

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

The first derivative of sqrt(x)(x^3-5x^2+1) is (7x^3-25x^2+1)/(2sqrt(x))

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

\frac{7 x ^{3} - 25 x ^{2} + 1}{2 x}

Solution steps

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

Apply the Product Rule:

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

f = x, g = x^{3} - 5 x^{2} + 1

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

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

\frac{d}{d x} (x^{3} - 5 x^{2} + 1) = 3 x^{2} - 10 x

= \frac{1}{2 x} (x^{3} - 5 x^{2} + 1) + (3 x^{2} - 10 x) x

Simplify

\frac{1}{2 x} (x^{3} - 5 x^{2} + 1) + (3 x^{2} - 10 x) x : \frac{7 x ^{3} - 25 x ^{2} + 1}{2 x}

= \frac{7 x ^{3} - 25 x ^{2} + 1}{2 x}

What is the derivative of sqrt(x)(x^3-5x^2+1) ?
The derivative of sqrt(x)(x^3-5x^2+1) is (7x^3-25x^2+1)/(2sqrt(x))
What is the first derivative of sqrt(x)(x^3-5x^2+1) ?
The first derivative of sqrt(x)(x^3-5x^2+1) is (7x^3-25x^2+1)/(2sqrt(x))