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Popular >

derivative of x^{5e^x}

Solution

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

Solution

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

Solution steps

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

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{5 e^{x}} = e^{5 e^{x} l n (x)}

= \frac{d}{d x} (e^{5 e^{x} l n (x)})

Apply the chain rule:

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

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

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

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

e^{5 e^{x} l n (x)} = x^{5 e^{x}}

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

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\int 2000 (x - 4) e^{x^{2} - 8 x} d x

Frequently Asked Questions (FAQ)

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