What is the derivative of e^xln(x^5) ?

The derivative of e^xln(x^5) is 5e^xln(x)+(5e^x)/x

What is the first derivative of e^xln(x^5) ?

The first derivative of e^xln(x^5) is 5e^xln(x)+(5e^x)/x

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

5 e^{x} ln (x) + \frac{5 e ^{x}}{x}

Solution steps

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

Apply the Product Rule:

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

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

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

\frac{d}{d x} (e^{x}) = e^{x}

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

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

Simplify

e^{x} ln (x^{5}) + \frac{5}{x} e^{x} : 5 e^{x} ln (x) + \frac{5 e ^{x}}{x}

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

What is the derivative of e^xln(x^5) ?
The derivative of e^xln(x^5) is 5e^xln(x)+(5e^x)/x
What is the first derivative of e^xln(x^5) ?
The first derivative of e^xln(x^5) is 5e^xln(x)+(5e^x)/x