9231 · 2.3
Differentiation
This topic gives you the tools to find the gradient of any point on a curve, even when the equation is tangled up (implicit) or described by a journey over time (parametric). It's about extending differentiation beyond simple 'y =...' functions.
Need to know
What you need to know
- **Differentiate term by term:** Apply the `d/dx` operator to every term in the equation.
- **Use the Chain Rule for y-terms:** Whenever you differentiate a term involving `y`, you must multiply by `dy/dx`. For example, `d/dx(sin(y)) = cos(y) * dy/dx`.
- **Use the Product Rule where necessary:** Be careful with terms like `xy` or `x²y³`. Remember to apply the product rule: `d/dx(uv) = u(dv/dx) + v(du/dx)`.
- **Solve for dy/dx:** After differentiating, rearrange the resulting equation algebraically to isolate `dy/dx`.
Explanation
Unlocking Complex Slopes
- For implicit functions (e.g., x² + y³ = 3), differentiate each term with respect to x, using the chain rule for y-terms (d/dx(y³) = 3y²(dy/dx)).
- Rearrange the resulting equation to make dy/dx the subject.
- For parametric functions (x=f(t), y=g(t)), find dy/dx using the formula: dy/dx = (dy/dt) / (dx/dt).
- To find the second derivative d²y/dx², differentiate your expression for dy/dx with respect to x, using the chain rule again: d/dx(dy/dx) = [d/dt(dy/dx)] * (dt/dx).