What is the derivative of x^2f(x^2+2x) ?

The derivative of x^2f(x^2+2x) is f(4x^3+6x^2)

What is the first derivative of x^2f(x^2+2x) ?

The first derivative of x^2f(x^2+2x) is f(4x^3+6x^2)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

derivative of

x^{2} f (x^{2} + 2 x)

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

Solution steps

\frac{d}{d x} (x^{2} f (x^{2} + 2 x))

Treat

f

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= f \frac{d}{d x} (x^{2} (x^{2} + 2 x))

Apply the Product Rule:

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

f = x^{2}, g = x^{2} + 2 x

= f (\frac{d}{d x} (x^{2}) (x^{2} + 2 x) + \frac{d}{d x} (x^{2} + 2 x) x^{2})

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

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

= f (2 x (x^{2} + 2 x) + (2 x + 2) x^{2})

Expand

2 x (x^{2} + 2 x) + (2 x + 2) x^{2} : 4 x^{3} + 6 x^{2}

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

What is the derivative of x^2f(x^2+2x) ?
The derivative of x^2f(x^2+2x) is f(4x^3+6x^2)
What is the first derivative of x^2f(x^2+2x) ?
The first derivative of x^2f(x^2+2x) is f(4x^3+6x^2)