Explained: "Find derivatives of the function defined as follows...." - Plainmath

coexpennan

Answered question

2021-10-04

Find derivatives of the function defined as follows.

y = 5 e^{- 4 x}

Answer & Explanation

unett

Skilled2021-10-05Added 119 answers

Step 1
To find the derivative of the function,

y = 5 e^{- 4 x}

Step 2
The derivative of the function is given by,

\frac{d y}{d x} = lim_{h \to 0} \frac{y (x + h) - y (x)}{h}

= lim_{h \to 0} \frac{5 e^{- 4 (x + h)} - 5 e^{- 4 x}}{h}

= lim_{h \to 0} \frac{5 e^{- 4 x} (e^{- 4 h} - 1)}{h}

= 5 e^{- 4 x} lim_{h \to 0} \frac{e^{- 4 h} - 1}{h}

= 5 e^{- 4 x} lim_{h \to 0} \frac{e^{- 4 h} - 1}{- 4 h} \times (- 4)

= 5 (- 4) e^{- 4 x} lim_{k \to 0} \frac{e^{k} - 1}{k}

where k=-4h as

h \to 0, k \to 0

= - 20 e^{- 4 x}

Step 3
Therefore the derivative of the function is -

\frac{d y}{d x} = - 20 e^{- 4 x}

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