Implicit Differentiation

Implicit Differentiation

by Thavarajah Selvarajah

​RECAP.......

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​LEVEL 1

Multiple Choice

Given  $$8x^3+10x+y^3-4y=0$$  .

$$24x^2+10+3y^2p-4p=0$$  .

What is  $$p?$$  

• $$1$$  

• $$\frac{\text{d}x}{\text{d}y}$$  

• $$\frac{dy}{dx}$$  

Multiple Choice

Given $$y^4+50=4xy$$  .

$$p=4xq+r$$  

What is p?

• $$4y^3$$  

• $$4y^3\ \frac{dy}{dx}$$  

• $$y^4$$  

• $$y^3$$  

Multiple Choice

Given $$y^4+50=4xy$$  .

$$p=4xq+r$$  

What is q?

• $$1$$  

• $$y$$  

• $$\frac{dy}{dx}$$  

• $$4x\ \frac{dy}{dx}$$  

Multiple Choice

Given $$y^4+50=4xy$$  .

$$p=4xq+r$$  

What is $$r$$  ?

• $$4xy$$  

• $$4y$$  

• $$1$$  

• $$4\ \frac{dy}{dx}$$  

Multiple Choice

Given  $$y\sin x\ +4\sec\ x\ =4$$  

$$yp+\frac{dy}{dx}q+4\sec x\tan x=0$$  .

What is p?

• $$1$$  

• $$\cos x$$  

• $$\sin x$$  

• $$-\sin x$$  

Multiple Choice

Given  $$y\sin x\ +4\sec\ x\ =4$$  

$$yp+\frac{dy}{dx}q+4\sec x\tan x=0$$  .

What is $$q$$  ?

• $$1$$  

• $$\cos x$$  

• $$\sin x$$  

• $$-\sin x$$  

Multiple Choice

Given  $$y\cot y=\cos2x+y$$  

$$p\cot y+yqp=r\sin2x+p$$  .

What is  $$p?$$  

• $$y$$  

• $$\frac{dy}{dx}$$  

• $$1$$  

Multiple Choice

Given  $$y\cot y=\cos2x+y$$  

$$p\cot y+yqp=r\sin2x+p$$  .

What is  $$r?$$  

• $$1$$  

• $$-1$$  

• $$2$$  

• $$-2$$  

Multiple Choice

Given  $$3\tan^2y+2x=2+x\cos y$$  .

$$6\left(\tan y\ \right)\ q\ p+2=\cos y+x\left(r\ p\right)$$  

What is  $$p?$$  

• $$\frac{\text{d}x}{\text{d}y}$$  

• $$\frac{dy}{dx}$$  

• $$1$$  

• $$-1$$  

Multiple Choice

Given  $$3\tan^2y+2x=2+x\cos y$$  .

$$6\left(\tan y\ \right)\ q\ p+2=\cos y+x\left(r\ p\right)$$  

What is  $$q?$$  

• $$\text{sec}^2y$$  

• $$\frac{dy}{dx}$$  

• $$1$$  

• $$\sec^2x$$  

Multiple Choice

Given  $$3\tan^2y+2x=2+x\cos y$$  .

$$6\left(\tan y\ \right)\ q\ p+2=\cos y+x\left(r\ p\right)$$  

What is  $$r?$$  

• $$\text{cos y}$$  

• $$\sin\ y$$  

• $$-\sin\ y$$  

• $$-\cos\ y$$  

​LEVEL 2

Multiple Choice

Given $$8x^3+10x+y^3-4y=0$$  . What is $$\frac{dy}{dx}=?$$  

• $$\frac{dy}{dx}=\frac{-24x^2-6}{3y^2}$$  

• $$\frac{dy}{dx}=\frac{-24x^2-10}{3y^2-4}$$  

• $$\frac{dy}{dx}=\frac{24x^2+10}{3y^3-4}$$  

• $$\frac{dy}{dx}=\frac{24x^2+6}{3y^2}$$  

Multiple Choice

Given $$y^4-4xy+50=0$$  . What is $$\frac{dy}{dx}=?$$  

• $$\frac{dy}{dx}=\frac{x}{y^3}$$  

• $$\frac{\text{d}y}{\text{d}x}=\frac{y}{y^3-x}$$  

• $$\frac{dy}{dx}=\frac{-y}{y^3-x}$$  

Multiple Choice

Given  $$y\sin x\ +4\sec\ x\ =4$$  . What is  $$\frac{dy}{dx}=?$$  

• $$\frac{dy}{dx}=\frac{-y\cos x-4\sec x\tan x\ }{\sin\ x}$$  

• $$\frac{dy}{dx}=\frac{-4\sec x\tan x}{\cos x}$$  

Multiple Choice

Given $$y\cot y=\cos\ 2x\ +y$$  . What is $$\frac{dy}{dx}=?$$  

• $$\frac{dy}{dx}=\frac{2\sin x}{\cot y+y\operatorname{cosec}^2y+1}$$

• $$\frac{dy}{dx}=\frac{-2\sin x}{\cot y-y\operatorname{cosec}^2y-1}$$  

• $$\frac{dy}{dx}=\frac{-\sin x}{\cot y-y\operatorname{cosec}^2y-1}$$

• $$\frac{dy}{dx}=\frac{-2\sin x}{\cot y+y\operatorname{cosec}^2y-1}$$

Multiple Choice

Given  $$3\ \tan^2y-x\cos y+2x=2$$  . What is  $$\frac{dy}{dx}=?$$  

• $$\frac{dy}{dx}=\frac{\cos y-2}{6\tan y\sec^2y+x\sin y}$$  

• $$\frac{dy}{dx}=\frac{\cos y-2}{6\tan y+x\sin y}$$  

• $$\frac{dy}{dx}=\frac{\cos y-2}{6\tan y\sec^2y-x\sin y}$$

• $$\frac{dy}{dx}=\frac{\cos y-2}{6\tan y-x\sin y}$$

Multiple Choice

Given $$e^{xy}+2y=4xy$$  .What is $$\frac{dy}{dx}=?$$  

• $$\frac{dy}{dx}=\frac{e^{xy}}{2}$$  

• $$\frac{dy}{dx}=\frac{ye^{xy}}{4x+2-xe^{xy}}$$