Derivatives and Differentiation Quiz

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Questions and Answers

What is the derivative of √8 + x√x?

  • 4√x + 1/√x
  • 3√x + √x
  • 2√x + 1/√x (correct)
  • 1/2√x + 1/2√x

For y = (x + 1)(x - 2), what is the correct derivative?

  • 2x + 3
  • 3x² + 2x - 2 (correct)
  • 3x² + 2x + 2
  • 3x² - 2x + 2

If ax + by = ab, what is the correct expression for dy/dx?

  • by (correct)
  • 2by
  • by (correct)
  • bx

What is the derivative of tan x with respect to cot x?

<p>sec x csc x (B), sec x csc x (D)</p> Signup and view all the answers

If y = tan⁻¹(cos 2x), what is dy/dx?

<p>-cos 2x (A)</p> Signup and view all the answers

What is the nth derivative of $2e^{x}$?

<p>$2ne^{x}$ (D)</p> Signup and view all the answers

If $xy=k^{2}$, what is y"?

<p>$\frac{2k^{2}}{x}$ (C)</p> Signup and view all the answers

What is the 3rd order derivative of $2^{x}$?

<p>$(\ln3)^{2}e^{x\ln2}$ (D)</p> Signup and view all the answers

For $f(x)=Ax+B\cos(x)$, what is the 2nd order derivative?

<p>None of these (B)</p> Signup and view all the answers

What is the nth derivative of $2e^{x}$?

<p>$2ne^{x}$ (B)</p> Signup and view all the answers

If $xy=k^{2}$, what is the second derivative of y with respect to x?

<p>$\frac{2k^{2}}{x^{3}}$ (C)</p> Signup and view all the answers

What is the 3rd order derivative of $2^{x}$?

<p>$(\ln3)^{2}e^{x\ln2}$ (C)</p> Signup and view all the answers

If $S$ represents the distance covered by a car, what does $\frac{d^{2}S}{dt^{2}}$ represent?

<p>Acceleration (A)</p> Signup and view all the answers

What is the nth derivative of $2e^{x}$?

<p>$2ne^{x}$ (B)</p> Signup and view all the answers

In the function $f(x)=Ax+B\cos(x)$, what is the 2nd order derivative?

<p>$(\ln 2)^3e^{x\ln 2}$ (B), $(\ln 2)^3e^{x\ln 2}$ (C), $(\ln 2)^3e^{x\ln 2}$ (D)</p> Signup and view all the answers

If $y=tan^{-1}x$, what is $y''(1)$?

<p>$\frac{-1}{4}$ (A)</p> Signup and view all the answers

What does $y=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...$ represent in terms of series?

<p>$\sin x$ (D)</p> Signup and view all the answers

If $S$ represents the distance covered by a car, what does $\frac{d^2S}{dt^2}$ represent?

<p>Deceleration (C)</p> Signup and view all the answers

What is the third order derivative of $2^{x}$?

<p>$(\ln 3)^2e^{x\ln 2}$ (A)</p> Signup and view all the answers

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Study Notes

Differentiation

  • The derivative of √8+ xvx is not provided.
  • If y=(x+1)(x-2), then y' = 2x - 1.

Second Derivative

  • If y=tan⁻¹x, then y''(1) = -1/2.

Higher Order Derivatives and Applications

  • If S is the distance covered by a car, then d²S/dt² is acceleration.

nth Derivative

  • The nth derivative of 2eˣ is 2neˣ.

Implicit Differentiation

  • If xy=k², then y" = 2k²/x³.

Applications of Derivatives

  • A function f(x) is said to be a decreasing function when x₁ > x₂ implies f(x₁) ≤ f(x₂).

Maclaurin Series

  • The Maclaurin series of sinx is x - x³/3! + x⁵/5! - ...

Derivatives of Trigonometric Functions

  • The derivative of tanx is sec²x.
  • The derivative of tanh ax is asechax.
  • The derivative of coth⁻¹(2x) is 2/(1 - 4x²).

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