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Popular >

derivative of x^a(1-x^b)

Solution

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

Solution

a x^{a - 1} (1 - x)^{b} - b x^{a} (1 - x)^{b - 1}

Solution steps

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

Apply the Product Rule:

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

f = x^{a}, g = (1 - x)^{b}

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

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

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

= a x^{a - 1} (1 - x)^{b} + (- b (1 - x)^{b - 1}) x^{a}

Simplify

= a x^{a - 1} (1 - x)^{b} - b x^{a} (1 - x)^{b - 1}

Popular Examples

limit as x approaches-8 of (-5x)/(x+8)

x \to - 8 lim (\frac{- 5 x}{x + 8})

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\int \frac{v}{0.122 - 0.0007 v ^{2}} d v

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\int \frac{9 x ^{2} - 49}{x ^{3}} d x

laplacetransform 7(t-7)laplace transform

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integral of (x^4+x+1+x^{-1}+x^{-2})

\int (x^{4} + x + 1 + x^{- 1} + x^{- 2}) d x

Frequently Asked Questions (FAQ)

What is the derivative of x^a(1-x^b) ?
The derivative of x^a(1-x^b) is ax^{a-1}(1-x)^b-bx^a(1-x)^{b-1}
What is the first derivative of x^a(1-x^b) ?
The first derivative of x^a(1-x^b) is ax^{a-1}(1-x)^b-bx^a(1-x)^{b-1}