Answer :
To find the derivative of the function [tex]\( f(x) = 9x^7 - 6x - 7 \)[/tex] using the definition of the derivative, we should understand that the derivative [tex]\( f'(x) \)[/tex] of a function [tex]\( f(x) \)[/tex] represents the rate at which [tex]\( f(x) \)[/tex] changes with respect to [tex]\( x \)[/tex].
For polynomial functions like this one, the derivative can be found using the power rule, which states that if [tex]\( f(x) = ax^n \)[/tex], then the derivative [tex]\( f'(x) \)[/tex] is [tex]\( anx^{n-1} \)[/tex].
Let's apply the power rule to each term in the function:
1. Differentiate [tex]\( 9x^7 \)[/tex]:
- Here, [tex]\( a = 9 \)[/tex] and [tex]\( n = 7 \)[/tex].
- Using the power rule: the derivative is [tex]\( 9 \times 7 \times x^{7-1} = 63x^6 \)[/tex].
2. Differentiate [tex]\(-6x\)[/tex]:
- Here, [tex]\( a = -6 \)[/tex] and [tex]\( n = 1 \)[/tex].
- Using the power rule: the derivative is [tex]\(-6 \times 1 \times x^{1-1} = -6\)[/tex].
3. Differentiate [tex]\(-7\)[/tex]:
- Since [tex]\(-7\)[/tex] is a constant, its derivative is [tex]\(0\)[/tex].
Now, combine all the derivatives:
- The derivative of the function [tex]\( f(x) = 9x^7 - 6x - 7 \)[/tex] is:
[tex]\[
f'(x) = 63x^6 - 6
\][/tex]
So, the derivative of the function is [tex]\( 63x^6 - 6 \)[/tex].
For polynomial functions like this one, the derivative can be found using the power rule, which states that if [tex]\( f(x) = ax^n \)[/tex], then the derivative [tex]\( f'(x) \)[/tex] is [tex]\( anx^{n-1} \)[/tex].
Let's apply the power rule to each term in the function:
1. Differentiate [tex]\( 9x^7 \)[/tex]:
- Here, [tex]\( a = 9 \)[/tex] and [tex]\( n = 7 \)[/tex].
- Using the power rule: the derivative is [tex]\( 9 \times 7 \times x^{7-1} = 63x^6 \)[/tex].
2. Differentiate [tex]\(-6x\)[/tex]:
- Here, [tex]\( a = -6 \)[/tex] and [tex]\( n = 1 \)[/tex].
- Using the power rule: the derivative is [tex]\(-6 \times 1 \times x^{1-1} = -6\)[/tex].
3. Differentiate [tex]\(-7\)[/tex]:
- Since [tex]\(-7\)[/tex] is a constant, its derivative is [tex]\(0\)[/tex].
Now, combine all the derivatives:
- The derivative of the function [tex]\( f(x) = 9x^7 - 6x - 7 \)[/tex] is:
[tex]\[
f'(x) = 63x^6 - 6
\][/tex]
So, the derivative of the function is [tex]\( 63x^6 - 6 \)[/tex].
