Explained: "Find derivatives of the functions defined as follows:..."

zgribestika

Answered question

2021-12-06

Find derivatives of the functions defined as follows:

y = (3 x^{3} - 4 x) e^{- 5 x}

mauricio0815sh

Beginner2021-12-07Added 34 answers

Step 1
Given:

y = (3 x^{3} - 4 x) e^{- 5 x}

Let:

y = f (x) = (3 x^{3} - 4 x) e^{- 5 x}

Product rule of differentiation is given by

\frac{d}{d x} (u \cdot v) = u \cdot \frac{d}{d x} (v) + v \cdot \frac{d}{d x} (u)

Here:

u = 3 x^{3} - 4 x

v = e^{- 5 x}

f^{'} (x) = (3 x^{3} - 4 x) \cdot \frac{d}{d x} (e^{- 5 x}) + (e^{- 5 x}) \cdot \frac{d}{d x} (3 x^{3} - 4 x)

Step 2

\frac{d}{d x} ((3 x^{3} - 4 x) e^{- 5 x})

= (3 x^{3} - 4 x) \cdot \frac{d}{d x} [e^{- 5 x}] + (e^{- 5 x}) \cdot \frac{d}{d x} [3 x^{3} - 4 x]

= (3 x^{3} - 4 x) e^{- 5 x} \cdot \frac{d}{d x} [- 5 x] + (e^{- 5 x}) (3 \frac{d}{d x} [x^{3}] - 4 \frac{d}{d x} [x])

= (3 x^{3} - 4 x) e^{- 5 x} (- 5 \frac{d}{d x} [x]) + (e^{- 5 x}) (3 \cdot 3 x^{2} - 4)

= - 5 (3 x^{3} - 4 x) e^{- 5 x} \cdot 1 + (e^{- 5 x}) (9 x^{2} - 4)

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