Resolved: "Find the derivative of f(x)=e−3x" - Plainmath

CheemnCatelvew

Answered question

2021-09-14

Find the derivative of

f (x) = e^{- 3 x}

Answer & Explanation

AGRFTr

Skilled2021-09-15Added 95 answers

Find the derivative of the following via implicit differentiation:

\frac{d}{d x} (f (x)) = \frac{d}{d x} (e^{- 3 x})

Using the chain rule,

\frac{d}{d x} (f (x)) = \frac{d f (u)}{d u} \frac{d u}{d x}

,
where

u = x and \frac{d}{d u} (f (u)) = f^{'} (u)

:

(\frac{d}{d x} (x)) f^{'} (x) = \frac{d}{d x} (e^{- 3 x})

The derivative of x is 1:

1 f^{'} (x) = \frac{d}{d x} (e^{- 3 x})

Using the chain rule,

\frac{d}{d x} (e^{- 3 x}) = \frac{d e^{u}}{d u} \frac{d u}{d x}, w h e r e u = - 3 x and \frac{d}{d u} (e^{u}) = e^{u}

:

f^{'} (x) = e^{- 3 x} (\frac{d}{d x} (- 3 x))

Factor out constants:

f^{'} (x) = - 3 (\frac{d}{d x} (x)) e^{- 3 x}

The derivative of x is 1:

f^{'} (x) = - 3 e^{- 3 x} 1

Simplify the expression:
Answer:

f^{'} (x) = - 3 e^{- 3 x}

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