Mathematics (AS)·Differentiation · NSSCAS 1.8.1
The gradient of a curve & differentiating xⁿ
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Welcome to calculus! In this lesson you will discover what the gradient of a curve really means — it is the gradient of the tangent at a point. You will meet the notations dy/dx, f′(x), d²y/dx² and f″(x), and learn the famous power rule: multiply by the power, knock the power down one. By the end you will differentiate whole-number powers, negative powers and square roots, and find the exact gradient of a curve at any point you choose.
What you'll learn in this lesson
By the end you should be able to (NSSCAS Mathematics (AS) 1.8.1):
- Interpret the gradient of a curve at a point as the gradient of the tangent at that point, and use the notations f′(x), f″(x), dy/dx and d²y/dx² (differentiation from first principles is not required).
- Use the derivative of xⁿ for any rational n, together with constant multiples, sums and differences of functions, to differentiate and to evaluate gradients and second derivatives.
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