Answer:
To find the derivative of the function \(f(x) = 9x^5 - 5x^4 + 8x - 6\), we'll apply the power rule for differentiation. The power rule states that if you have a term of the form \(ax^n\), where \(a\) is a constant and \(n\) is a real number, then its derivative is \(anx^{(n-1)}\).
Let's find the derivative of each term in your function:
1. For the term \(9x^5\), apply the power rule:
\[ \frac{d}{dx}(9x^5) = 9 \cdot 5x^{5-1} = 45x^4\]
2. For the term \(-5x^4\), apply the power rule:
\[ \frac{d}{dx}(-5x^4) = -5 \cdot 4x^{4-1} = -20x^3\]
3. For the term \(8x\), apply the power rule (with \(n = 1\)):
\[ \frac{d}{dx}(8x) = 8 \cdot 1x^{1-1} = 8\]
4. The constant term \(-6\) does not have an \(x\) variable, so its derivative is zero.
Now, add up the derivatives of all the terms to find the derivative of the entire function \(f(x)\):
\[ f'(x) = 45x^4 - 20x^3 + 8 \]
So, the derivative of \(f(x) = 9x^5 - 5x^4 + 8x - 6\) is \(f'(x) = 45x^4 - 20x^3 + 8\).
Explanation: