What is the derivative of 4xe^{-kx} ?

The derivative of 4xe^{-kx} is 4(e^{-xk}-e^{-xk}xk)

What is the first derivative of 4xe^{-kx} ?

The first derivative of 4xe^{-kx} is 4(e^{-xk}-e^{-xk}xk)

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derivative of

4 x e^{- k x}

4 (e^{- x k} - e^{- x k} x k)

Solution steps

\frac{d}{d x} (4 x e^{- k x})

Treat

k

as a constant

Take the constant out:

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

= 4 \frac{d}{d x} (x e^{- k x})

Apply the Product Rule:

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

f = x, g = e^{- k x}

= 4 (\frac{d x}{d x} e^{- k x} + \frac{d}{d x} (e^{- k x}) x)

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

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

= 4 (1 \cdot e^{- k x} + (- k e^{- k x}) x)

Simplify

= 4 (e^{- x k} - e^{- x k} x k)

What is the derivative of 4xe^{-kx} ?
The derivative of 4xe^{-kx} is 4(e^{-xk}-e^{-xk}xk)
What is the first derivative of 4xe^{-kx} ?
The first derivative of 4xe^{-kx} is 4(e^{-xk}-e^{-xk}xk)