1. **State the problem:** Find the derivative of the function $$f(x) = 4x^5 - 3x^3 + 2x^2 - 7x + 5$$. 2. **Recall the power rule for derivatives:** For any term of the form $$ax^n$$, the derivative is $$a n x^{n-1}$$. 3. **Apply the power rule to each term:** - Derivative of $$4x^5$$ is $$4 \times 5 x^{5-1} = 20x^4$$. - Derivative of $$-3x^3$$ is $$-3 \times 3 x^{3-1} = -9x^2$$. - Derivative of $$2x^2$$ is $$2 \times 2 x^{2-1} = 4x$$. - Derivative of $$-7x$$ is $$-7 \times 1 x^{1-1} = -7$$. - Derivative of constant $$5$$ is $$0$$. 4. **Combine all derivatives:** $$f'(x) = 20x^4 - 9x^2 + 4x - 7$$. 5. **Final answer:** $$\boxed{f'(x) = 20x^4 - 9x^2 + 4x - 7}$$