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Chapter 4: Problem 29

Differentiate. $$ f(x)=e^{-x^{2}+7 x} $$

Short Answer

Expert verified
The derivative is \( f'(x) = e^{-x^2 + 7x} \times (-2x + 7) \).

Step by step solution

01

Identify the Function to Differentiate

The function given is \[ f(x) = e^{-x^2 + 7x}. \]We need to differentiate this function with respect to \( x \).
02

Apply the Chain Rule

The outer function is \( e^{u} \) and the inner function is \( u = -x^2 + 7x \). According to the chain rule, \[ f'(x) = \frac{d}{dx}(e^{u}) \times \frac{d}{dx}(u). \]
03

Differentiate the Outer Function

The derivative of the outer function \( e^{u} \) with respect to \( u \) is \[ \frac{d}{du} (e^u) = e^u. \]
04

Differentiate the Inner Function

Differentiate the inner function \( u = -x^2 + 7x \) with respect to \( x \). So we get: \[ \frac{d}{dx} (-x^2 + 7x) = -2x + 7. \]
05

Combine the Results from the Chain Rule

Apply the chain rule results: \[ f'(x) = e^{u} \times (-2x + 7). \]Substitute back the value of \( u = -x^2 + 7x \): \[ f'(x) = e^{-x^2 + 7x} \times (-2x + 7). \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

chain rule
The chain rule is a crucial concept in calculus for differentiating composite functions. When you have a function inside another function, the chain rule helps you find the rate of change. Let's break it down:
Imagine you have a function of the form \( f(g(x)) \). The chain rule states:
  • First, differentiate the outer function as if the inner function was just a variable. So, if you have \( f(u) \) and \( u = g(x)\), differentiate \( f(u) \). In mathematical terms, this is \[ \frac{d}{dx} f(g(x)) = f'(g(x)) \].

  • Then, multiply this result by the derivative of the inner function \( g(x) \).

For our exercise, the outer function is \( e^u \), and the inner function is \( u = -x^2 + 7x \). Applying the chain rule: \[ f'(x) = \frac{d}{dx}(e^{u}) \times \frac{d}{dx}(u) \]. This simplifies the differentiation process for more complex problems.
derivative of exponential function
The derivative of an exponential function is surprisingly straightforward. For an exponential function \( e^{u} \), where \( u \) is any function of \( x \), the differentiation rule is:
  • The derivative of \( e^{u} \) with respect to \( u \) is the same: \[ \frac{d}{du} (e^u) = e^u \].
This makes it easier to work with exponential functions since you don't have to change it much, just remember to multiply by the derivative of \( u \).
In our specific exercise, we first identified \( u = -x^2 + 7x \).
So, differentiating \( e^{u} \) with respect to \( x \) becomes: \[ f'(x) = e^{u} \times \frac{d}{dx}(u) \]. This keeps the exponential function in its original form but acknowledges the inner function's rate of change.
differentiation of polynomial functions
Differentiating polynomial functions is a fundamental skill in calculus. Polynomial functions are just sums of terms that look like \( ax^n \). The differentiation rule is simple:
  • The power rule: \[ \frac{d}{dx} (ax^n) = anx^{(n-1)} \].
For our exercise, we use this rule to differentiate \(-x^2 + 7x \):
  • The term \(-x^2 \) becomes \[ -2x \].
  • The term \(7x \) becomes \[ 7 \].

Combining these, the derivative of the inner function is: \[ \frac{d}{dx} (-x^2 + 7x) = -2x + 7 \].
Finally, applying this to our chain rule, we get the complete derivative: \[ f'(x) = e^{-x^2 + 7x} \times (-2x + 7) \]. This shows how both rules integrate to solve a more complex differentiation problem.

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