التفاضل (نقي 3)

The derivative 导数 that completes the circle

• You know the derivatives of $\sin$, $\cos$, $\tan$, $e^x$, and $\ln x$. Pure 3 adds one more: the inverse tangent 反正切.
• Together, these standard derivatives let you differentiate any combination of elementary functions.

Review of Pure 2 methods

• The methods are those of Pure 2: the product, quotient and chain rules 链式法则, with parametric 参数 and implicit curves.
• These are the workhorses of differentiation — master them and you can handle any problem.

z on an Argand diagram: modulus is the distance, argument the angle, conjugate the reflection

The inverse tangent derivative

• One new standard derivative:

• This appears in integration (the reverse) and in related rates problems.

Implicit differentiation 隐函数微分 revisited

• For equations like $x^2 + y^2 = 25$, differentiate every term with respect to $x$.
• When you meet a $y$ term, apply the chain rule: $\dfrac{d}{dx}(y^2) = 2y\dfrac{dy}{dx}$.
• Then solve for $\dfrac{dy}{dx}$.

allow_no_figure: Review lesson consolidating Pure 2 differentiation methods (product, quotient, chain rules) with one new derivative — no new visual concepts introduced.

Worked example — implicit differentiation

• Find $\dfrac{dy}{dx}$ for $x^2 + xy + y^2 = 7$.
• Differentiate: $2x + y + x\dfrac{dy}{dx} + 2y\dfrac{dy}{dx} = 0$.
• Collect $\dfrac{dy}{dx}$ terms: $(x + 2y)\dfrac{dy}{dx} = -(2x + y)$.
• $\dfrac{dy}{dx} = -\dfrac{2x + y}{x + 2y}$.
• Differentiate relations given implicitly or parametrically.

You've got it