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derivative of (y(x^3e^{2x}+1))

Solution

\frac{d}{d x} ((y) (x^{3} e^{2 x} + 1))

Solution

\frac{d y}{d x} (x^{3} e^{2 x} + 1) + (2 e^{2 x} x^{3} + 3 e^{2 x} x^{2}) y

Solution steps

\frac{d}{d x} ((y) (x^{3} e^{2 x} + 1))

Apply the Product Rule:

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

f = y, g = x^{3} e^{2 x} + 1

= \frac{d y}{d x} (x^{3} e^{2 x} + 1) + \frac{d}{d x} (x^{3} e^{2 x} + 1) y

\frac{d}{d x} (x^{3} e^{2 x} + 1) = 2 e^{2 x} x^{3} + 3 e^{2 x} x^{2}

= \frac{d y}{d x} (x^{3} e^{2 x} + 1) + (2 e^{2 x} x^{3} + 3 e^{2 x} x^{2}) y

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Frequently Asked Questions (FAQ)

What is the derivative of (y(x^3e^{2x}+1)) ?
The derivative of (y(x^3e^{2x}+1)) is (dy)/(dx)(x^3e^{2x}+1)+(2e^{2x}x^3+3e^{2x}x^2)y
What is the first derivative of (y(x^3e^{2x}+1)) ?
The first derivative of (y(x^3e^{2x}+1)) is (dy)/(dx)(x^3e^{2x}+1)+(2e^{2x}x^3+3e^{2x}x^2)y