Derivative e to the power of minus x

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The derivative of the exponential function is the exponential function itself. Unfortunately, this simple rule does not apply to compound exponentials such as e to the power of minus x. Here you need the chain rule.

You need the chain rule.
You need the chain rule.

What you need:

  • Basic concepts of derivation rules

Chain rule for derivatives - simply explained

  • The chain rule is for Derivatives from Functions responsible, which are referred to as composite. They can (mostly) be recognized by the fact that another function is "hidden" in a function.
  • Examples of such functions are sin (x²) or e-x³. In both cases two functions are linked, namely x² in the angle function sin and -x³ as the exponent of the exponential function.
  • To derive such functions, you need the hidden function as an auxiliary function as well as the output function and its derivatives.
  • According to the chain rule, it is true that the derivative of the original function is equal to the derivative of the output function times the derivative of the auxiliary function. Sounds complicated, but it isn't, as the example "e to the power of minus x" will show in a moment.

Derive e to the power of minus x - that's how it's done

mathematics write the common form f (x) = e for "e to the power of minus x"-x. You are looking for the derivation of this function.

Math - the chain rule and its application simply explained

In mathematics there are different ways to derive a function ...

  1. First, you need to realize that -x is the hidden function here. You take this as an auxiliary function, it is simply referred to as z = -x (in some mathematics works this auxiliary function is also referred to as g (x); However, z is easier to use, like point 2. shows).
  2. The (simplified) output function is then f (z) = ez.
  3. For the chain rule you still need the derivatives of the two functions. We have z '= -1 (the derivative of -x is -1) and f' (z) = ez (The derivative of the exponential function is the exponential function itself, only the argument is now z).
  4. According to the chain rule, the derivative of the total function is obtained by multiplying the two derivatives f '(z) and z'. So you get f '(x) = f' (z) * z '= ez * (-1) = - ez = - e-x. Please note that you have to use the auxiliary function z again, after all, the variable of f (x) is x and not z.

So the derivative of "e to the power of minus x" is simply "-e to the power of minus x".

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