What is the derivative of x^8ln(5x) ?

The derivative of x^8ln(5x) is 8x^7ln(5x)+x^7

What is the first derivative of x^8ln(5x) ?

The first derivative of x^8ln(5x) is 8x^7ln(5x)+x^7

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\frac{d}{d x} (x^{8} ln (5 x))

8 x^{7} ln (5 x) + x^{7}

Solution steps

\frac{d}{d x} (x^{8} ln (5 x))

Apply the Product Rule:

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

f = x^{8}, g = ln (5 x)

= \frac{d}{d x} (x^{8}) ln (5 x) + \frac{d}{d x} (ln (5 x)) x^{8}

\frac{d}{d x} (x^{8}) = 8 x^{7}

\frac{d}{d x} (ln (5 x)) = \frac{1}{x}

= 8 x^{7} ln (5 x) + \frac{1}{x} x^{8}

\frac{1}{x} x^{8} = x^{7}

= 8 x^{7} ln (5 x) + x^{7}

What is the derivative of x^8ln(5x) ?
The derivative of x^8ln(5x) is 8x^7ln(5x)+x^7
What is the first derivative of x^8ln(5x) ?
The first derivative of x^8ln(5x) is 8x^7ln(5x)+x^7