Continuity and Differentiability — Part 2: Inverse trigonometric, implicit, logarithmic, parametric and second derivatives: visual lesson, CBSE Class 12 Mathematics

Lesson

Continuity and Differentiability, Part 2

Five techniques for functions that are not written simply as $y = f(x)$ .

In this lesson you will

Differentiate inverse trigonometric functions
Differentiate implicitly defined functions
Differentiate exponential and logarithmic functions and use logarithmic differentiation
Differentiate functions given in parametric form
Find second-order derivatives and verify relations involving them

Slope $x^x(1 + \log x)$ is $0$ at $x = \dfrac{1}{e}$, the lowest point M.

Concept

Differentiate $\sin^{-1} x$ , $\cos^{-1} x$ , $\tan^{-1} x$ and their compositions
Find $\dfrac{dy}{dx}$ when $x$ and $y$ are mixed in one equation
Differentiate $e^x$ , $\log x$ and use logarithmic differentiation
Differentiate parametric curves
Find $\dfrac{d^2y}{dx^2}$ and prove relations

Concept

Within their principal ranges these functions are differentiable, with: