## 19 Questions

What is the derivative of √8 + x√x?

2√x + 1/√x

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

3x² + 2x - 2

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

by

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

sec x csc x

If y = tan⁻¹(cos 2x), what is dy/dx?

-cos 2x

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

$2ne^{x}$

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

$\frac{2k^{2}}{x}$

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

$(\ln3)^{2}e^{x\ln2}$

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

None of these

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

$2ne^{x}$

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

$\frac{2k^{2}}{x^{3}}$

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

$(\ln3)^{2}e^{x\ln2}$

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

Acceleration

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

$2ne^{x}$

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

$(\ln 2)^3e^{x\ln 2}$

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

$\frac{-1}{4}$

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

$\sin x$

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

Deceleration

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

$(\ln 3)^2e^{x\ln 2}$

## 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).

Test your knowledge of derivatives and differentiation with this quiz. Select the correct options related to finding derivatives of various functions and expressions. Questions include product rule, chain rule, and trigonometric functions.

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