What is the derivative of (cos(x2x^3)/(e^x)) ?

The derivative of (cos(x2x^3)/(e^x)) is (2x^2(-xsin(x)+3cos(x)-xcos(x)))/(e^x)

What is the first derivative of (cos(x2x^3)/(e^x)) ?

The first derivative of (cos(x2x^3)/(e^x)) is (2x^2(-xsin(x)+3cos(x)-xcos(x)))/(e^x)

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\frac{d}{d x} (\frac{cos ( x ) 2 x ^{3}}{e ^{x}})

\frac{2 x ^{2} ( - x sin ( x ) + 3 cos ( x ) - x cos ( x ) )}{e ^{x}}

Solution steps

\frac{d}{d x} (\frac{cos ( x ) \cdot 2 x ^{3}}{e ^{x}})

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= 2 \cdot \frac{\frac{d}{d x} ( cos ( x ) x ^{3} ) e ^{x} - \frac{d}{d x} ( e ^{x} ) cos ( x ) x ^{3}}{( e ^{x} ) ^{2}}

\frac{d}{d x} (cos (x) x^{3}) = - x^{3} sin (x) + 3 x^{2} cos (x)

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

= 2 \cdot \frac{( - x ^{3} sin ( x ) + 3 x ^{2} cos ( x ) ) e ^{x} - e ^{x} cos ( x ) x ^{3}}{( e ^{x} ) ^{2}}

Simplify

2 \cdot \frac{( - x ^{3} sin ( x ) + 3 x ^{2} cos ( x ) ) e ^{x} - e ^{x} cos ( x ) x ^{3}}{( e ^{x} ) ^{2}} : \frac{2 x ^{2} ( - x sin ( x ) + 3 cos ( x ) - x cos ( x ) )}{e ^{x}}

= \frac{2 x ^{2} ( - x sin ( x ) + 3 cos ( x ) - x cos ( x ) )}{e ^{x}}

What is the derivative of (cos(x2x^3)/(e^x)) ?
The derivative of (cos(x2x^3)/(e^x)) is (2x^2(-xsin(x)+3cos(x)-xcos(x)))/(e^x)
What is the first derivative of (cos(x2x^3)/(e^x)) ?
The first derivative of (cos(x2x^3)/(e^x)) is (2x^2(-xsin(x)+3cos(x)-xcos(x)))/(e^x)