What is the derivative of ax^{a-1}ln(x+x^{a-1}) ?

The derivative of ax^{a-1}ln(x+x^{a-1}) is a^2x^{a-2}ln(x)-ax^{a-2}ln(x)+2ax^{a-2}-x^{a-2}

What is the first derivative of ax^{a-1}ln(x+x^{a-1}) ?

The first derivative of ax^{a-1}ln(x+x^{a-1}) is a^2x^{a-2}ln(x)-ax^{a-2}ln(x)+2ax^{a-2}-x^{a-2}

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\frac{d}{d x} (a x^{a - 1} ln (x) + x^{a - 1})

a^{2} x^{a - 2} ln (x) - a x^{a - 2} ln (x) + 2 a x^{a - 2} - x^{a - 2}

Solution steps

\frac{d}{d x} (a x^{a - 1} ln (x) + x^{a - 1})

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (a x^{a - 1} ln (x)) + \frac{d}{d x} (x^{a - 1})

\frac{d}{d x} (a x^{a - 1} ln (x)) = a (x^{a - 2} ln (x) (a - 1) + x^{a - 2})

\frac{d}{d x} (x^{a - 1}) = x^{a - 2} (a - 1)

= a (x^{a - 2} ln (x) (a - 1) + x^{a - 2}) + x^{a - 2} (a - 1)

Simplify

a (x^{a - 2} ln (x) (a - 1) + x^{a - 2}) + x^{a - 2} (a - 1) : a^{2} x^{a - 2} ln (x) - a x^{a - 2} ln (x) + 2 a x^{a - 2} - x^{a - 2}

= a^{2} x^{a - 2} ln (x) - a x^{a - 2} ln (x) + 2 a x^{a - 2} - x^{a - 2}

- \frac{6 e ^{\frac{6}{x}}}{x ^{2}}

\frac{d}{d x} (\frac{5 x ^{2}}{2 b ^{2} + x ^{2}})

(y^{2} + x y) d x + x^{2} d y = 0

20 y^{^{''}} + 11 y^{^{'}} - 3 y = 0

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

What is the derivative of ax^{a-1}ln(x+x^{a-1}) ?
The derivative of ax^{a-1}ln(x+x^{a-1}) is a^2x^{a-2}ln(x)-ax^{a-2}ln(x)+2ax^{a-2}-x^{a-2}
What is the first derivative of ax^{a-1}ln(x+x^{a-1}) ?
The first derivative of ax^{a-1}ln(x+x^{a-1}) is a^2x^{a-2}ln(x)-ax^{a-2}ln(x)+2ax^{a-2}-x^{a-2}