What is the derivative of y=x^4cos(4x^2) ?

The derivative of y=x^4cos(4x^2) is 4x^3cos(4x^2)-8x^5sin(4x^2)

What is the first derivative of y=x^4cos(4x^2) ?

The first derivative of y=x^4cos(4x^2) is 4x^3cos(4x^2)-8x^5sin(4x^2)

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derivative of

y = x^{4} cos (4 x^{2})

4 x^{3} cos (4 x^{2}) - 8 x^{5} sin (4 x^{2})

Solution steps

\frac{d}{d x} (x^{4} cos (4 x^{2}))

Apply the Product Rule:

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

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

= \frac{d}{d x} (x^{4}) cos (4 x^{2}) + \frac{d}{d x} (cos (4 x^{2})) x^{4}

\frac{d}{d x} (x^{4}) = 4 x^{3}

\frac{d}{d x} (cos (4 x^{2})) = - sin (4 x^{2}) \cdot 8 x

= 4 x^{3} cos (4 x^{2}) + (- sin (4 x^{2}) \cdot 8 x) x^{4}

Simplify

4 x^{3} cos (4 x^{2}) + (- sin (4 x^{2}) \cdot 8 x) x^{4} : 4 x^{3} cos (4 x^{2}) - 8 x^{5} sin (4 x^{2})

= 4 x^{3} cos (4 x^{2}) - 8 x^{5} sin (4 x^{2})

What is the derivative of y=x^4cos(4x^2) ?
The derivative of y=x^4cos(4x^2) is 4x^3cos(4x^2)-8x^5sin(4x^2)
What is the first derivative of y=x^4cos(4x^2) ?
The first derivative of y=x^4cos(4x^2) is 4x^3cos(4x^2)-8x^5sin(4x^2)