Answer

**Differentiation of a √x **

When we differentiate the function √x or x^(1/2), we use the power rule of differentiation.

Let y = √x or x^(1/2) , then we can rewrite y as y = x^(1/2).

To find the derivative of y with respect to x, we use the power rule which states that if y = x^n, then the derivative dy/dx = nx^(n-1).

Applying this rule, we have y' = 1/2 \cdot x^{- \frac{1}{2}}.

Therefore, the derivative of √x or x^{\frac{1}{2}}is 1/(2√x) or 1/(2x^\frac{1}{2}).

This means that the rate of change of the function √x is inversely proportional to 2 times the square root of x.

You can also differentiate coefficient of x in the following form

**Common functions**

7

6x

x^{1/2}

2x^2

**Exponential functions**

-e^{-\frac{12}{7x}}

-3e^{2x}

**Trigonometric functions**

4 \sin (x)

-\cos(3x)

**Logarithmic functions**

4In9x

**1. Find the derivative of the function -2x^{-\frac{1}{2}}.**

**Solution**:

To find the derivative of a function in the form f(x) = ax^n, where a and n are constants, we use the power rule. The derivative is given by:

f'(x) = n \cdot a \cdot x^{n-1}.

Applying this rule to the given function, we have:

f'(x) = -\frac{1}{2} \cdot -2 \cdot x^{-\frac{1}{2}-1}</p><p>= x^{-\frac{3}{2}}.

Therefore, the derivative of -2x^{-\frac{1}{2}} is

x^{-\frac{3}{2}}.

**2. Determine the derivative of e^{x}.**

**Solution:**

The derivative of e^{x} is itself, as the derivative of

e^{x} is e^{x}.

This is a unique property of the exponential function e^{x}.

Therefore, the derivative of e^{x} is

e^{x}.

**3. Calculate the derivative of 4\cos(9x).**

**Solution:**

By applying the chain rule and derivative of cosine function, we have:

f'(x) = -4 \cdot 9 \sin(9x) = -36 \sin(9x).

Therefore, the derivative of 4\cos(9x) is

-36 \sin(9x).

**4. Find the derivative of -3e^{\frac{8}{5}x}.**

**Solution:**

Similarly to the derivative of e^{x}, the derivative of e^{\frac{8}{5}x} is itself times the constant \frac{8}{5}:

f'(x) = -3 \cdot \frac{8}{5} e^{\frac{8}{5}x} = -\frac{24}{5} e^{\frac{8}{5}x}.

Therefore, the derivative of -3e^{\frac{8}{5}x} is

-\frac{24}{5} e^{\frac{8}{5}x}.

**5. Differentiate the function 4\sin(2x).**

**Solution:**

Using the chain rule and derivative of the sine function, we get:

f'(x) = 4 \cdot 2 \cos(2x)= 8 \cos(2x).

Therefore, the derivative of 4\sin(2x) is

8\cos(2x).

1. \frac{d}{dx}\left(-2x^{-\frac{1}{2}}\right) = x^{-\frac{3}{2}}

2. \frac{d}{dx}\left(-e^x\right) = -e^x

3. \frac{d}{dx}\left(4\cos(9x)\right) = -36\sin(9x)

4. \frac{d}{dx}\left(-3e^{\frac{8}{5}x}\right) = -\frac{24}{5}e^{\frac{8}{5}x}

5. \frac{d}{dx}\left(5\ln x\right) = \frac{5}{x}

6. \frac{d}{dx}\left(2x^3\right) = 6x^2

7. \frac{d}{dx}\left(9e^{-2x}\right) = -18e^{-2x}

8. \frac{d}{dx}\left(7\sin(2x)\right) = 14\cos(2x)

9. \frac{d}{dx}\left(-4e^{5x}\right) = -20e^{5x}

10. \frac{d}{dx}\left(3\ln(4x)\right) = \frac{3}{x}

11. \frac{d}{dx}\left(-5\cos(3x)\right) = 15\sin(3x)

12. \frac{d}{dx}\left(6e^{-4x}\right) = -24e^{-4x}

13. \frac{d}{dx}\left(2\sin(5x)\right) = 10\cos(5x)

14. \frac{d}{dx}\left(8e^{2x}\right) = 16e^{2x}

15. \frac{d}{dx}\left(4\ln(3x)\right) = \frac{4}{x}

For best result write

1 as **1**

-1 as **-1**

x as **x**

-x as **-x**

x^{1/2} as **x^(1/2)**

x^{-1/2} as **x^-(1/2)**

-x^{1/2} as** - x^(1/2)**

-x^{-1/2} as **- x^-(1/2)**

-2x as -2x

2x^2 as 2x^2

-2x^2 as -2x^2

2x^{1/2} as 2x^(1/2)

2x^{-1/2} as 2x^-(1/2)

-2x^{1/2} as -2x^(1/2)

-2x^{-1/2} as -2x^-(1/2)

2x^{-1} as 2x^-1

-2x^{-1} as -2x^-1

-x^{-1} as -x^-1

x^{-1} as x^-1

x^2 as x^2

-x^2 -x^2

-x^{-2} -x^-2

2x as 2x

e^x as e^x

-e^x as -e^x

-e^{-x} as -e^-x

e^{2x} as e^(2x)

e^{-2x} as e^-(2x)

-e^{2x} as -e^-(2x)

-e^{-2x} as e^-(2x)

e^{8/5x} as e^(8/5x)

e^{-8/5x} as e^-(8/5x)

-e^{8/5x} as -e^-(8/5x)

-e^{-8/5x} as -e^-(8/5x)

3e^x as 3e^x

3e^{-x} as 3e^-x

-3e^x as -3e^x

-3e^{-x} as -3e^-x

3e^{2x} as 3e^(2x)

3e^{-2x} as 3e^-(2x)

-3e^{2x} as -3e^(2x)

-3e^{-2x} as -3e^(-2x)

3e^{8/5x} as 3e^(8/5x)

3e^{-8/5x} as 3e^(-8/5x)

-3e^{8/5x} as -3e^(8/5x)

-3e^{-8/5x} as -3e^(-8/5x)

-\cos(x) as -cos(x)

4\sin(x) as 4 sin(x)

-\tan(3x) - tan(3x)

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Answer

Answer

Order of differentiation

1

2

3

4

5

6

7

8

9