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1. To determine the first and second derivatives of the function f(x) = 5/x – 9x + 4, we begin by rewriting the function in a more manageable form. The term 5/x can be expressed as 5x^(-1), allowing us to apply the power rule for differentiation more easily. Thus, the function can be rewritten as f(x) = 5x^(-1) – 9x + 4. This transformation facilitates the differentiation process, as we can now differentiate each term individually.
2. The first derivative, denoted as f'(x), is obtained by applying the power rule to each term of the rewritten function. For the term 5x^(-1), the derivative is -5x^(-2), which can also be expressed as -5/x^2. The derivative of -9x is simply -9, and the constant term 4 has a derivative of 0. Therefore, combining these results, we find that the first derivative of the function is f'(x) = -5/x^2 – 9. This calculation is essential for students engaging in Christmas A Level Maths Revision, as it reinforces the application of differentiation techniques.
3. To find the second derivative, denoted as f”(x), we differentiate the first derivative f'(x) = -5/x^2 – 9 once more. Applying the power rule again, we rewrite -5/x^2 as -5x^(-2). The derivative of -5x^(-2) is 10x^(-3), or 10/x^3, while the derivative of the constant -9 remains 0. Consequently, the second derivative is f”(x) = 10/x^3. This process of finding both the first and second derivatives is crucial for understanding the behaviour of the function and is a fundamental aspect of calculus that students must master.
