Questions: Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating. 35. f(x)=4-3x^7

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.
35. f(x)=4-3x^7

Transcript text: Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating. 35. $f(x)=4-3 x^{7}$

Solution

Solution Steps

To find the derivative of the function $ f(x) = 4 - 3x^7 $, we will apply the power rule of differentiation. The power rule states that the derivative of $ x^n $ is $ nx^{n-1} $. We will differentiate each term separately and then combine the results.

Step 1: Identify the Function

The function given is $ f(x) = 4 - 3x^7 $.

Step 2: Apply the Power Rule

To find the derivative, apply the power rule: if $ f(x) = ax^n $, then $ f'(x) = anx^{n-1} $.

The derivative of $ 4 $ is $ 0 $ since it is a constant.
The derivative of $ -3x^7 $ is $ -21x^6 $ because $ -3 \times 7 = -21 $ and $ x^{7-1} = x^6 $.

Step 3: Combine the Derivatives

Combine the derivatives of each term:

\[ f'(x) = 0 - 21x^6 = -21x^6 \]