1. The problem is to find the derivative of the function $f(x) = 3x^2 + 5x - 7$. 2. The formula for the derivative of a polynomial function $f(x) = ax^n$ is $f'(x) = n \cdot a x^{n-1}$. 3. Applying the power rule to each term: - Derivative of $3x^2$ is $2 \cdot 3 x^{2-1} = 6x$. - Derivative of $5x$ is $1 \cdot 5 x^{1-1} = 5$. - Derivative of constant $-7$ is $0$. 4. Combining these results, the derivative is: $$f'(x) = 6x + 5$$ 5. This derivative tells us the slope of the tangent line to the curve at any point $x$.