Unit 3: Differentiation: Composite, Implicit, and Inverse Differentiation

This unit is where derivatives become flexible. Instead of only differentiating simple formulas, we learn how derivatives behave under composition, implicit relationships, and inverse functions.

Chain rule

If \(y = f(g(x))\), then

\[\frac{dy}{dx} = f'(g(x))g'(x).\]

Examples:

\[\frac{d}{dx} (3x^2+1)^5 = 5(3x^2+1)^4(6x)\] \[\frac{d}{dx} \sin(x^2) = \cos(x^2)(2x)\]

Implicit differentiation

When a curve is defined by an equation relating \(x\) and \(y\), differentiate both sides with respect to \(x\) and remember that \(y\) depends on \(x\).

Example:

\[x^2 + y^2 = 25\]

Differentiate:

\[2x + 2y\frac{dy}{dx} = 0\]

so

\[\frac{dy}{dx} = -\frac{x}{y}.\]

[Image Placeholder: circle with tangent line showing slope found implicitly]

Derivatives of inverse functions

If \(f\) is differentiable and invertible with \(f'(a) \ne 0\), then

\[(f^{-1})'(b) = \frac{1}{f'(a)}\]

where \(b = f(a)\).

Equivalent formula:

\[(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}.\]

Derivatives of inverse trig functions

\[\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}\] \[\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}\] \[\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}\]

Chain-rule forms:

\[\frac{d}{dx}\arcsin(u) = \frac{u'}{\sqrt{1-u^2}}, \qquad \frac{d}{dx}\arctan(u) = \frac{u'}{1+u^2}.\]

Exponential and logarithmic differentiation

Useful rules:

\[\frac{d}{dx} e^{u(x)} = e^{u(x)}u'(x)\] \[\frac{d}{dx} \ln(\lvert u(x) \rvert) = \frac{u'(x)}{u(x)}\]

Logarithmic differentiation is especially helpful when powers and products are mixed or when both base and exponent contain variables.

Example:

If

\[y = x^x,\]

then

\[\ln y = x\ln x\]

and differentiating gives

\[y' = x^x(\ln x + 1).\]

Strategy:

  1. Draw and label a diagram.
  2. Write an equation relating the variables.
  3. Differentiate implicitly with respect to time.
  4. Substitute the requested instant.
  5. Keep units consistent.

[Image Placeholder: ladder against wall with changing x and y distances]

Advanced chain-rule patterns

You should be comfortable stacking rules:

  • product + chain,
  • quotient + chain,
  • trig + chain,
  • inverse trig + chain,
  • exponential/log + chain.

Parametric preview

Later, if

\[x = f(t), \qquad y = g(t),\]

then

\[\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\]

when \(dx/dt \ne 0\). This is the parametric analogue of the chain rule.

Common mistakes

  • Forgetting to multiply by the derivative of the inside.
  • Treating \(y\) as a constant during implicit differentiation.
  • Dropping the factor \(dy/dx\).
  • Using inverse notation incorrectly: \(\sin^{-1}x\) means \(\arcsin x\), not \(1/\sin x\).
Made by Joshua Bie