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

The derivative of f(x)=xcos(x^2) is cos(x^2)-2x^2sin(x^2)

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

The first derivative of f(x)=xcos(x^2) is cos(x^2)-2x^2sin(x^2)

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derivative of

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

cos (x^{2}) - 2 x^{2} sin (x^{2})

Solution steps

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

Apply the Product Rule:

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

f = x, g = cos (x^{2})

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

\frac{d x}{d x} = 1

\frac{d}{d x} (cos (x^{2})) = - sin (x^{2}) \cdot 2 x

= 1 \cdot cos (x^{2}) + (- sin (x^{2}) \cdot 2 x) x

Simplify

1 \cdot cos (x^{2}) + (- sin (x^{2}) \cdot 2 x) x : cos (x^{2}) - 2 x^{2} sin (x^{2})

= cos (x^{2}) - 2 x^{2} sin (x^{2})

What is the derivative of f(x)=xcos(x^2) ?
The derivative of f(x)=xcos(x^2) is cos(x^2)-2x^2sin(x^2)
What is the first derivative of f(x)=xcos(x^2) ?
The first derivative of f(x)=xcos(x^2) is cos(x^2)-2x^2sin(x^2)