Implicit Differentiation — Syllabus Points

1. First Derivative $\frac{dy}{dx}$

• 1.1 Differentiate equations involving $x$ and $y$ implicitly
• 1.2 Apply chain rule: $\frac{d}{dx}f(y)=f'(y)\frac{dy}{dx}$
• 1.3 Apply product rule to terms like $xy$, $x^2y$, $x\sin y$
• 1.4 Rearrange to express $\frac{dy}{dx}$ as an algebraic fraction

2. Second Derivative $\frac{d^2y}{dx^2}$

• 2.1 Differentiate the expression for $\frac{dy}{dx}$ implicitly again
• 2.2 Substitute the original equation to simplify
• 2.3 Express $\frac{d^2y}{dx^2}$ in terms of $x$ and $y$ only

3. Values at Specific Points

• 3.1 Substitute coordinates into $\frac{dy}{dx}$ to find gradient at a point
• 3.2 Find equations of tangents and normals
• 3.3 Determine stationary points ($\frac{dy}{dx}=0$) from implicit equations

4. Combined Techniques

• 4.1 Use logarithmic differentiation for $y=f(x)^{g(x)}$ type functions
• 4.2 Combine implicit differentiation with parametric forms