To find the derivative [tex]\( f'(x) \)[/tex] of the function [tex]\( f(x) = 7x^8 + 5x^6 - 3x^5 + 2x^4 - 1 \)[/tex], we need to apply the power rule of differentiation. The power rule states that if [tex]\( f(x) = x^n \)[/tex], then [tex]\( f'(x) = nx^{n-1} \)[/tex].
Let's differentiate each term of the polynomial function separately:
1. For the term [tex]\( 7x^8 \)[/tex]:
[tex]\[
\frac{d}{dx}(7x^8) = 7 \cdot 8 x^{8-1} = 56x^7
\][/tex]
2. For the term [tex]\( 5x^6 \)[/tex]:
[tex]\[
\frac{d}{dx}(5x^6) = 5 \cdot 6 x^{6-1} = 30x^5
\][/tex]
3. For the term [tex]\( -3x^5 \)[/tex]:
[tex]\[
\frac{d}{dx}(-3x^5) = -3 \cdot 5 x^{5-1} = -15x^4
\][/tex]
4. For the term [tex]\( 2x^4 \)[/tex]:
[tex]\[
\frac{d}{dx}(2x^4) = 2 \cdot 4 x^{4-1} = 8x^3
\][/tex]
5. For the constant term [tex]\( -1 \)[/tex]:
[tex]\[
\frac{d}{dx}(-1) = 0
\][/tex]
Now, we sum up all these derivatives to get [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(x) = 56x^7 + 30x^5 - 15x^4 + 8x^3
\][/tex]
Therefore, the derivative of the function [tex]\( f(x) \)[/tex] is:
[tex]\[
f'(x) = 56x^7 + 30x^5 - 15x^4 + 8x^3
\][/tex]
