What is (\partial)/(\partial x)((x^3+x^4)(x^2+3x)^2) ?

The answer to (\partial)/(\partial x)((x^3+x^4)(x^2+3x)^2) is 8x^7+49x^6+90x^5+45x^4

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(\partial)/(\partial x)((x^3+x^4)(x^2+3x)^2)

\frac{\partial}{\partial x} ((x^{3} + x^{4}) (x^{2} + 3 x)^{2})

8 x^{7} + 49 x^{6} + 90 x^{5} + 45 x^{4}

Solution steps

\frac{\partial}{\partial x} ((x^{3} + x^{4}) (x^{2} + 3 x)^{2})

Apply the Product Rule:

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

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

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

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

\frac{\partial}{\partial x} ((x^{2} + 3 x)^{2}) = 2 (x^{2} + 3 x) (2 x + 3)

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

Simplify

(3 x^{2} + 4 x^{3}) (x^{2} + 3 x)^{2} + 2 (x^{2} + 3 x) (2 x + 3) (x^{3} + x^{4}) : 8 x^{7} + 49 x^{6} + 90 x^{5} + 45 x^{4}

= 8 x^{7} + 49 x^{6} + 90 x^{5} + 45 x^{4}

\int_{1}^{4} (5 x^{3} - 6) d x

\frac{d y}{d x} = 5 y e^{x + 8}

\int sec^{3} (7 x) d x

y^{^{'}} = sec (2) (x - y)

f (1) = 2 x^{4} - x^{3} + 6 x^{2}

What is (\partial)/(\partial x)((x^3+x^4)(x^2+3x)^2) ?
The answer to (\partial)/(\partial x)((x^3+x^4)(x^2+3x)^2) is 8x^7+49x^6+90x^5+45x^4