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Popular >

derivative of x^pe^{ax}

Solution

\frac{d}{d x} (x^{p} e^{a x})

Solution

x^{p} (ln (x) \frac{d p}{d x} + \frac{p}{x}) e^{a x} + e^{a x} a x^{p}

Solution steps

\frac{d}{d x} (x^{p} e^{a x})

Apply the Product Rule:

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

f = x^{p}, g = e^{a x}

= \frac{d}{d x} (x^{p}) e^{a x} + \frac{d}{d x} (e^{a x}) x^{p}

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

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

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

Popular Examples

integral of y/(1-y^2)

\int \frac{y}{1 - y ^{2}} d y

(\partial)/(\partial x)(-1+ln(x))

\frac{\partial}{\partial x} (- 1 + ln (x))

integral of 1/(\sqrt[4]{8-3x)}

derivative of (4x+x^2-2)/((2+x)^2)derivative of

\frac{4 x + x ^{2} - 2}{( 2 + x ) ^{2}}

derivative of (3x^2-5/(2x^2+1))

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

Frequently Asked Questions (FAQ)

What is the derivative of x^pe^{ax} ?
The derivative of x^pe^{ax} is x^p(ln(x)(dp)/(dx)+p/x)e^{ax}+e^{ax}ax^p
What is the first derivative of x^pe^{ax} ?
The first derivative of x^pe^{ax} is x^p(ln(x)(dp)/(dx)+p/x)e^{ax}+e^{ax}ax^p