What is the derivative of (y(x^3+ax^2+bx+c)) ?

The derivative of (y(x^3+ax^2+bx+c)) is (dy)/(dx)(x^3+ax^2+bx+c)+(3x^2+2ax+b)y

What is the first derivative of (y(x^3+ax^2+bx+c)) ?

The first derivative of (y(x^3+ax^2+bx+c)) is (dy)/(dx)(x^3+ax^2+bx+c)+(3x^2+2ax+b)y

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\frac{d}{d x} ((y) (x^{3} + a x^{2} + b x + c))

\frac{d y}{d x} (x^{3} + a x^{2} + b x + c) + (3 x^{2} + 2 a x + b) y

Solution steps

\frac{d}{d x} ((y) (x^{3} + a x^{2} + b x + c))

Apply the Product Rule:

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

f = y, g = x^{3} + a x^{2} + b x + c

= \frac{d y}{d x} (x^{3} + a x^{2} + b x + c) + \frac{d}{d x} (x^{3} + a x^{2} + b x + c) y

\frac{d}{d x} (x^{3} + a x^{2} + b x + c) = 3 x^{2} + 2 a x + b

= \frac{d y}{d x} (x^{3} + a x^{2} + b x + c) + (3 x^{2} + 2 a x + b) y

x + h \cdot (- x)

x^{3} y d x + (x^{4} + x^{2} y^{2} + y^{4}) d y

4 (\frac{y ^{2}}{8} - y + 2)

(2 cos (x) - 2) (2 sin (x) - 2)

\frac{x}{( x - 4 ) ( x + 2 )}

What is the derivative of (y(x^3+ax^2+bx+c)) ?
The derivative of (y(x^3+ax^2+bx+c)) is (dy)/(dx)(x^3+ax^2+bx+c)+(3x^2+2ax+b)y
What is the first derivative of (y(x^3+ax^2+bx+c)) ?
The first derivative of (y(x^3+ax^2+bx+c)) is (dy)/(dx)(x^3+ax^2+bx+c)+(3x^2+2ax+b)y