What is the derivative of ln(sqrt(x)(x^2-5)) ?

The derivative of ln(sqrt(x)(x^2-5)) is (x^2-5)/(2x)+xln(x)

What is the first derivative of ln(sqrt(x)(x^2-5)) ?

The first derivative of ln(sqrt(x)(x^2-5)) is (x^2-5)/(2x)+xln(x)

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\frac{d}{d x} (ln (x) (x^{2} - 5))

\frac{x ^{2} - 5}{2 x} + x ln (x)

Solution steps

\frac{d}{d x} (ln (x) (x^{2} - 5))

Apply the Product Rule:

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

f = ln (x), g = x^{2} - 5

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

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

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

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

Simplify

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

= \frac{x ^{2} - 5}{2 x} + x ln (x)

What is the derivative of ln(sqrt(x)(x^2-5)) ?
The derivative of ln(sqrt(x)(x^2-5)) is (x^2-5)/(2x)+xln(x)
What is the first derivative of ln(sqrt(x)(x^2-5)) ?
The first derivative of ln(sqrt(x)(x^2-5)) is (x^2-5)/(2x)+xln(x)