Implicit, Parametric and Logarithmic Differentiation
Implicit differentiation differentiates an equation term by term treating y as a function of x; parametric form uses dy/dx = (dy/dt)/(dx/dt); logarithmic differentiation handles variable exponents.
What is Implicit, Parametric and Logarithmic Differentiation?
Implicit differentiation differentiates an equation term by term treating y as a function of x; parametric form uses dy/dx = (dy/dt)/(dx/dt); logarithmic differentiation handles variable exponents.
Key formula / rule: Every y term contributes a factor dy/dx by the chain rule.
Key points
- Compute dy/dx for an implicitly defined relation.
- Apply logarithmic differentiation to variable-exponent functions.
Common exam trap
Treating y as a constant while differentiating an implicit relation.
Definitions
- Term
Implicit, Parametric and Logarithmic Differentiation
- Meaning
Implicit differentiation differentiates an equation term by term treating y as a function of x; parametric form uses dy/dx = (dy/dt)/(dx/dt); logarithmic differentiation handles variable exponents.
- Term
Implicit, Parametric and Logarithmic Differentiation — explanation
- Meaning
When y cannot be isolated, differentiating the relation still yields the slope. Logarithmic differentiation converts products, quotients and powers into sums, which is far safer for expressions such as xx.
Learning objectives
Compute dy/dx for an implicitly defined relation.
Apply logarithmic differentiation to variable-exponent functions.
Formulae
- Key point
Every y term contributes a factor dy/dx by the chain rule.
- Key point
For parametric curves dy/dx = (dy/dt)/(dx/dt) provided dx/dt is non-zero.
- Key point
Take logarithms before differentiating when the exponent contains x.
Prerequisites
MAT-U2-LC-T2-S1-C1
Common mistakes
Treating y as a constant while differentiating an implicit relation.
Writing the second derivative of a parametric curve as (d2y/dt2)/(d2x/dt2).
Keywords
Implicit
Parametric
Logarithmic
Differentiation
Practice preview
If x = at^2 and y = 2at, find dy/dx.…
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If y = (sin x)^x + x^(sin x), find dy/dx.…
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