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Popular Calculus >

derivative of f(x)=x^3(x-4)^2

Solution

derivative of

f (x) = x^{3} (x - 4)^{2}

Solution

5 x^{4} - 32 x^{3} + 48 x^{2}

Solution steps

\frac{d}{d x} (x^{3} (x - 4)^{2})

Apply the Product Rule:

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

f = x^{3}, g = (x - 4)^{2}

= \frac{d}{d x} (x^{3}) (x - 4)^{2} + \frac{d}{d x} ((x - 4)^{2}) x^{3}

\frac{d}{d x} (x^{3}) = 3 x^{2}

\frac{d}{d x} ((x - 4)^{2}) = 2 (x - 4)

= 3 x^{2} (x - 4)^{2} + 2 (x - 4) x^{3}

Simplify

3 x^{2} (x - 4)^{2} + 2 (x - 4) x^{3} : 5 x^{4} - 32 x^{3} + 48 x^{2}

= 5 x^{4} - 32 x^{3} + 48 x^{2}

Graph

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Frequently Asked Questions (FAQ)

What is the derivative of f(x)=x^3(x-4)^2 ?
The derivative of f(x)=x^3(x-4)^2 is 5x^4-32x^3+48x^2
What is the first derivative of f(x)=x^3(x-4)^2 ?
The first derivative of f(x)=x^3(x-4)^2 is 5x^4-32x^3+48x^2