UpStudy Free Solution:

To find the derivative of \(y = e^ { 5- 4x} \) with respect to \(x\), we use the chain rule.

The chain rule states that if \(y = e^ { u( x) } \), then

\[\frac { dy} { dx} = e^ { u( x) } \cdot \frac { du} { dx} \]

Here, \(u( x) = 5 - 4x\).

First, we find the derivative of \(u( x) \) with respect to \(x\):

\[\frac { du} { dx} = \frac { d} { dx} ( 5 - 4x) = - 4\]

Now, apply the chain rule:

\[\frac { dy} { dx} = e^ { 5- 4x} \cdot ( - 4) \]

Therefore, the derivative is:

\[\frac { dy} { dx} = - 4 e^ { 5- 4x} \]

So the correct answer is:

\[- 4 e^ { 5- 4x} \]

Supplemental Knowledge

To find the derivative of a function, we apply differentiation rules. For exponential functions of the form \(y = e^ { u( x) } \), where \(u( x) \) is a function of \(x\), the derivative with respect to \(x\) is given by:

\[\frac { dy} { dx} = e^ { u( x) } \cdot \frac { du} { dx} \]

Here, we use the chain rule. The chain rule states that if a variable \(y\) depends on an intermediate variable \(u\), which in turn depends on \(x\), then the derivative of \(y\) with respect to \(x\) is:

\[\frac { dy} { dx} = \frac { dy} { du} \cdot \frac { du} { dx} \]

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