Differentiation

Differentiation

Dr. Dhaval Thakkar

Multiple Choice

Derivative means the same thing as

• slope of the tangent line

• slope of the normal line

• exponent

• potato

Multiple Choice

The derivative of a function is its

• Sign

• Maximum/Minimum

• Instantaneous rate of change

• Common Denominator

Multiple Choice

If a function has a derivative that is negative, what does that tell you?

• The function is increasing

• The function is decreasing

Multiple Choice

• 3x²

• x³

• 6x

• 3x² + x³

Multiple Choice

What is the derivative of y = 2π?

• 0

• 2

• -2

• None of the above

Multiple Choice

What is the derivative of xⁿ?

• (n-1)xⁿ

• nxⁿ⁺¹

• (n+1)x^n-1

• nx^n-1

Multiple Choice

Differentiate with respect to x...

• 6x² + 6x

• 6x + 2

• 6x + 6

• 6x² + 6

Multiple Choice

Differentiate with respect to x...

• 6x² + 6x

• 6x + 2

• 6x + 6

• 6x² + 6

Multiple Choice

At which point(s) will the slopes of the tangent line are zero?

• at C only

• at points A, C and E only

• at point B and D only

• at points A and E only

Multiple Choice

What rule best apply to find the derivative of y=x lnx

• product rule

• quotient rule

• power rule

• sum rule

Multiple Choice

What is the derivative of f?

• 1/(x - 1)^-1

• -1/(x - 1)^-2

• -1/(x - 1)

• -1/(x - 1)²

Multiple Choice

Find the derivative:

y=5x²e^3x

• y'=10xe^3x(2x+3)

• y'=5xe^3x(3x+2)

• y'=10ex^3x(3x+2)

• y'=5xe^3x(2x+3)

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(e^{\left(2x+1\right)}\right)^2$  

•  $4e^{\left(4x+2\right)}$  

•  $\frac{4}{e^{\left(4x+2\right)}}$  

•  $e^{\left(4x+2\right)}\left(8x^2+2x\right)$  

•  $\frac{\left(8x^2+2x\right)}{e^{\left(4x+2\right)}}$  

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(e^x+2x\right)^3$  

•  $3\left(e^x+2\right)\left(e^x+2x\right)^2$  

•  $3\left(e^x+2\right)^2$  

•  $3\left(e^x+4x^2\right)\left(e^x+2x\right)^2$  

•  $4\left(e^x+2\right)^4$  

Multiple Choice

Find the derivative of

• sinx

• -sinx

• -cscx cotx

• sec²x

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(\sin x\right)$  

•  $\cos x$  

•  $-\cos x$  

•  $\sin^{-1}x$  

•  $\frac{1}{\sin x}$  

•  $\sin x\cos x$  

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(\sec x\right)$  

•  $\sec x\tan x$  

•  $-\sec x\tan x$  

•  $\sec^{-1}x$  

•  $\frac{1}{\sec x}$  

•  $\csc x\cot x$  

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(\csc x\right)$  

•  $-\csc x\cot x$  

•  $\csc x\cot x$  

•  $\csc^{-1}x\cot^{-1}x$  

•  $\frac{1}{\csc x\cot x}$  

•  $-\sec x\tan x$  

Multiple Choice

Find the derivative of

• cscx cotx

• secx tanx

• sec²x

• -cscx cotx

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(\cot x\right)$  

•  $-\csc^2x$  

•  $\csc^2x$  

•  $\cot^{-1}x$  

•  $\frac{1}{\cot x}$  

•  $\csc x$  

Multiple Choice

The derivative of sin(2x)

• 2cos(x)

• 2sin(2x)

• 2cos(2x)

• 2xcos(2x)

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(3\tan^4x\right)$  

•  $12\tan^3x\sec^2x$  

•  $12\tan^3x\sec x$  

•  $12\sec^3x$  

•  $12\sec^6x$  

•  $12\sec^2x$  

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(\frac{\sin x}{x^2}\right)$  

•  $\frac{x\cos x-2\sin x}{x^3}$  

•  $-\frac{x\cos x+2\sin x}{x^3}$  

•  $\frac{x\cos x+2\sin x}{x^3}$  

•  $\frac{-x\cos x+2\sin x}{x^3}$  

•  $\frac{x\cos x-2\sin x}{x^4}$  

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(2\cos\left(2x\right)+4\sin\left(2x+3\right)\right)$  

•  $-4\sin\left(2x\right)+8\cos\left(2x+3\right)$  

•  $4\sin\left(2x\right)+8\cos\left(2x+3\right)$  

•  $-4\sin\left(2x\right)-8\cos\left(2x+3\right)$  

•  $4\sin\left(2x\right)-8\cos\left(2x+3\right)$  

•  $-4\sin\left(2x\right)\cos\left(2x\right)+8\sin\left(2x+3\right)\cos\left(2x+3\right)$  

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(\sqrt{3\sin x+4\cos x}\right)$  

•  $\frac{3\cos x-4\sin x}{2\sqrt{3\sin x+4\cos x}}$  

•  $\frac{3\cos x+4\sin x}{2\sqrt{3\sin x+4\cos x}}$  

•  $\frac{3}{2}\left(3\cos x-4\sin x\right)\sqrt{\left(3\sin x+4\cos x\right)^3}$  

•  $\frac{3}{2}\left(3\cos x+4\sin x\right)\sqrt{\left(3\sin x+4\cos x\right)^3}$  

•  $-\frac{3\cos x-4\sin x}{2\sqrt{3\sin x+4\cos x}}$  

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(\frac{x}{\cos x}\right)^2$  

•  $\frac{2x\left(\cos x+x\sin x\right)}{\cos^3x}$  

•  $\frac{2x\left(\cos x-x\sin x\right)}{\cos^3x}$  

•  $\frac{2x\left(\cos^2x+x\sin x\right)}{\cos^4x}$  

•  $\frac{2x\left(\cos^2x-x\sin x\right)}{\cos^4x}$  

•  $-\frac{2x\left(\cos x+x\sin x\right)}{\cos^3x}$  

Multiple Choice

 $\frac{\text{d}}{\text{d}y}\left(2\sin x+x^3\right)$  

•  $2\cos x+3x^2$  

•  $-2\cos x+3x^2$  

•  $2\cos x+4x^4$  

•  $2\cos x+2x^2$  

•  $-2\cos x+2x^2$  

Multiple Choice

Find  $\frac{dy}{dx}$  for  $y=\ln\left(\frac{2x+1}{1-2x}\right)$  .

•  $\frac{2}{2x+1}-\frac{2}{1-2x}$  

•  $\frac{2}{2x+1}+\frac{2}{1-2x}$  

•  $\frac{2}{2x-1}+\frac{2}{1-2x}$  

•  $\frac{2}{2x+1}+\frac{2}{1+2x}$  

Multiple Choice

Find dy/dx by Implicit Differentiation 
x³ +y³  = 36

• 6 -x
• 3x² +3y² 
• −x²/y²
• 0

Multiple Choice

If

 $y=x^x\ then\ find\ \frac{\text{d}y}{\text{d}x}$  

•  $x^x\left(1+\ln x\right)$  

•  $x^x\left(\ln x\right)$  

•  $x.x^{x-1}$  

•  $\left(1+\ln x\right)$  

Multiple Choice

Find dy/dx by Implicit Differentiation 
x + 4y = 1

• -1/4
• -1 
• 4
• 1