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Popular >

derivative of |x|^x

Solution

\frac{d}{d x} (∣ x ∣^{x})

Solution

∣ x ∣^{x} (ln ∣ x ∣ + 1)

Solution steps

\frac{d}{d x} (∣ x ∣^{x})

Apply exponent rule:

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

∣ x ∣^{x} = e^{x l n ∣ x ∣}

= \frac{d}{d x} (e^{x l n ∣ x ∣})

Apply the chain rule:

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

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

\frac{d}{d x} (x ln ∣ x ∣) = ln ∣ x ∣ + 1

= e^{x l n ∣ x ∣} (ln ∣ x ∣ + 1)

e^{x l n ∣ x ∣} = ∣ x ∣^{x}

= ∣ x ∣^{x} (ln ∣ x ∣ + 1)

Popular Examples

integral of 1/(x-sqrt(x-2))

\int \frac{1}{x - x - 2} d x

(dy)/(dx)=((3x+2y))/(3x+2y+2)

\frac{d y}{d x} = \frac{( 3 x + 2 y )}{3 x + 2 y + 2}

(\partial)/(\partial x)(x^4+x*sin(y))

\frac{\partial}{\partial x} (x^{4} + x \cdot sin (y))

integral of 1/(x(x-2))

\int \frac{1}{x ( x - 2 )} d x

integral of (6400q)/(16q^2+3200)

\int \frac{6400 q}{16 q ^{2} + 3200} d q

Frequently Asked Questions (FAQ)

What is the derivative of |x|^x ?
The derivative of |x|^x is |x|^x(ln|x|+1)
What is the first derivative of |x|^x ?
The first derivative of |x|^x is |x|^x(ln|x|+1)