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Popular >

derivative of x^ay^b

Solution

\frac{d}{d x} (x^{a} y^{b})

Solution

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

Solution steps

\frac{d}{d x} (x^{a} y^{b})

Apply the Product Rule:

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

f = x^{a}, g = y^{b}

= \frac{d}{d x} (x^{a}) y^{b} + \frac{d}{d x} (y^{b}) x^{a}

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

\frac{d}{d x} (y^{b}) = b y (x)^{b - 1} \frac{d y}{d x}

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

Popular Examples

derivative of (1/(x^2)-5/(x^4))(x+8x^3)derivative of

(\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) (x + 8 x^{3})

integral of 7x^3(x-5)^{-4}

\int 7 x^{3} (x - 5)^{- 4} d x

derivative of (30/((1-x)^4))

\frac{d}{d x} (\frac{30}{( 1 - x ) ^{4}})

y^{'''}+4y^{''}-6y^'-12y=0

y^{^{'''}} + 4 y^{^{''}} - 6 y^{^{'}} - 12 y = 0

5y^{''}+y^'=-2x

5 y^{^{''}} + y^{^{'}} = - 2 x

Frequently Asked Questions (FAQ)

What is the derivative of x^ay^b ?
The derivative of x^ay^b is ax^{a-1}y^b+by(x)^{b-1}(dy)/(dx)x^a
What is the first derivative of x^ay^b ?
The first derivative of x^ay^b is ax^{a-1}y^b+by(x)^{b-1}(dy)/(dx)x^a