Introductions to Derivatives

Derivatives help us understand how a function changes as its input changes. They measure the rate of change and are crucial in fields like physics, economics, and machine learning. By understanding derivatives, we can analyze trends, optimize processes, and predict behavior.

The Limit Definition of a Derivative

The derivative of a function $f(x)$ at a specific point $x = a$ is given by:

\lim_{h \rarr 0} \frac{f(x + h) - f(x)}{h}

This formula tells us how much $f(x)$ changes when we make a tiny step $h$ along the x-axis. The smaller $h$ becomes, the closer we get to the instantaneous rate of change.

Basic Derivative Rules

Power Rule

If a function is a power of $x$ , the derivative follows:

\frac{d}{dx}x^n=nx^{n-1}

This means that when differentiating , we bring the exponent down and reduce it by one:

\frac{d}{dx}x^3=3x^2

Constant Rule

The derivative of any constant is zero:

\frac{d}{dx}C=0

For example, if $f(x) = 5$ , then:

\frac{d}{dx}5=0

Sum & Difference Rule

The derivative of a sum or difference of functions follows:

\frac{d}{dx} \left[ f(x) \pm g(x) \right] = f'(x) \pm g'(x)

For example, differentiating separately:

\frac{d}{dx}(x^3 + 2x) = 3x^2 + 2

Product & Quotient Rules

Product Rule

If two functions are multiplied, the derivative is found as follows:

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

This means we differentiate each function separately and then sum their products. If $f(x)=x^2$ and $g(x) = e^x$ , then:

\frac{d}{dx}[x^2e^x] = 2xe^x + x^3e^x

Quotient Rule

When dividing functions, use:

\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}

If $f(x)=x^2$ and $g(x)=x+1$ , then:

\frac{d}{dx} \left[ \frac{x^2}{x + 1} \right] = \frac{2x(x+1) - x^2(1)}{(x+1)^2}

Chain Rule: Differentiating Composite Functions

When differentiating nested functions, use:

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

For example, if $y = (3x + 2)^5$ , then:

\frac{d}{dx}(3x+2)^5 = 5(3x+2)^4 \cdot 3 = 15(3x+2)^4

This rule is essential in neural networks and machine learning algorithms.

Exponential Chain Rule Example:

When you're differentiating something like:

y =e^{2x^2}

You're dealing with a composite function:

Outer function: $e^u$
Inner function: $u = 2x^2$

Apply the chain rule step-by-step:

\frac{d}{dx}2x^2=4x

Then multiply by the original exponential:

\frac{d}{dx}\left( e^{2x^2} \right) = 4x \cdot e^{2x^2}

Study More

In machine learning and neural nets, this shows up when working with exponential activations or loss functions.

Logarithmic Chain Rule Example:

Let's differentiate $\ln(2x)$ . Again, it's a composite function — log on the outside, linear on the inside.

Differentiate the inner part:

\frac{d}{dx}(2x)=2

Now apply the chain rule to the log:

\frac{d}{dx}\ln(2x) = \frac{1}{2x} \cdot 2

Which simplifies to:

\frac{d}{dx}\ln(2x) = \frac{2}{2x} = \frac{1}{x}

Note

Even if you’re differentiating $ln(kx)$ , the result is always $1/x$ because the constants cancel out.

Special Case: Derivative of the Sigmoid Function

The sigmoid function is commonly used in machine learning:

\sigma(x) = \frac{1}{1+x^{-x}}

Its derivative plays a key role in optimization:

\sigma'(x) = \sigma(x)(1 - \sigma(x))

If $f(x) = \frac{1}{1 + e^{-x}}$ , then:

f'(x) = \frac{e^{-x}}{(1 + e^{-x})^2}

This formula ensures that gradients remain smooth during training.

1. Which of the following correctly represents the derivative of $x^4$ ?

2. The derivative of a constant is always:

3. What is the derivative of $x^5$ ?

4. Given the following sigmoid function:

g(x) = \frac{2}{1+e^{-x}}

What is its derivative $g'(x)$ ?

Which of the following correctly represents the derivative of $x^4$ ?

Select the correct answer

4x^3

x^3

4x^4

3x^2

The derivative of a constant is always:

Select the correct answer

$1$

$C$

Undefined

$0$

What is the derivative of $x^5$ ?

Select the correct answer

x^4

$5x^4$

5x^5

4x^3

Given the following sigmoid function:

g(x) = \frac{2}{1+e^{-x}}

What is its derivative $g'(x)$ ?

Select the correct answer

g(x) + 2

g(x)^2

g(x)(2 - g(x))

2e^{-x}

War alles klar?

Wie können wir es verbessern?

Danke für Ihr Feedback!

Abschnitt 3. Kapitel 3

Fragen Sie AI

Fragen Sie alles oder probieren Sie eine der vorgeschlagenen Fragen, um unser Gespräch zu beginnen

Awesome!

Completion rate improved to 1.89

Introductions to Derivatives

Swipe um das Menü anzuzeigen

The Limit Definition of a Derivative

The derivative of a function $f(x)$ at a specific point $x = a$ is given by:

\lim_{h \rarr 0} \frac{f(x + h) - f(x)}{h}

This formula tells us how much $f(x)$ changes when we make a tiny step $h$ along the x-axis. The smaller $h$ becomes, the closer we get to the instantaneous rate of change.

Basic Derivative Rules

Power Rule

If a function is a power of $x$ , the derivative follows:

\frac{d}{dx}x^n=nx^{n-1}

This means that when differentiating , we bring the exponent down and reduce it by one:

\frac{d}{dx}x^3=3x^2

Constant Rule

The derivative of any constant is zero:

\frac{d}{dx}C=0

For example, if $f(x) = 5$ , then:

\frac{d}{dx}5=0

Sum & Difference Rule

The derivative of a sum or difference of functions follows:

\frac{d}{dx} \left[ f(x) \pm g(x) \right] = f'(x) \pm g'(x)

For example, differentiating separately:

\frac{d}{dx}(x^3 + 2x) = 3x^2 + 2

Product & Quotient Rules

Product Rule

If two functions are multiplied, the derivative is found as follows:

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

This means we differentiate each function separately and then sum their products. If $f(x)=x^2$ and $g(x) = e^x$ , then:

\frac{d}{dx}[x^2e^x] = 2xe^x + x^3e^x

Quotient Rule

When dividing functions, use:

\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}

If $f(x)=x^2$ and $g(x)=x+1$ , then:

\frac{d}{dx} \left[ \frac{x^2}{x + 1} \right] = \frac{2x(x+1) - x^2(1)}{(x+1)^2}

Chain Rule: Differentiating Composite Functions

When differentiating nested functions, use:

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

For example, if $y = (3x + 2)^5$ , then:

\frac{d}{dx}(3x+2)^5 = 5(3x+2)^4 \cdot 3 = 15(3x+2)^4

This rule is essential in neural networks and machine learning algorithms.

Exponential Chain Rule Example:

When you're differentiating something like:

y =e^{2x^2}

You're dealing with a composite function:

Outer function: $e^u$
Inner function: $u = 2x^2$

Apply the chain rule step-by-step:

\frac{d}{dx}2x^2=4x

Then multiply by the original exponential:

\frac{d}{dx}\left( e^{2x^2} \right) = 4x \cdot e^{2x^2}

Study More

In machine learning and neural nets, this shows up when working with exponential activations or loss functions.

Logarithmic Chain Rule Example:

Let's differentiate $\ln(2x)$ . Again, it's a composite function — log on the outside, linear on the inside.

Differentiate the inner part:

\frac{d}{dx}(2x)=2

Now apply the chain rule to the log:

\frac{d}{dx}\ln(2x) = \frac{1}{2x} \cdot 2

Which simplifies to:

\frac{d}{dx}\ln(2x) = \frac{2}{2x} = \frac{1}{x}

Note

Even if you’re differentiating $ln(kx)$ , the result is always $1/x$ because the constants cancel out.

Special Case: Derivative of the Sigmoid Function

The sigmoid function is commonly used in machine learning:

\sigma(x) = \frac{1}{1+x^{-x}}

Its derivative plays a key role in optimization:

\sigma'(x) = \sigma(x)(1 - \sigma(x))

If $f(x) = \frac{1}{1 + e^{-x}}$ , then:

f'(x) = \frac{e^{-x}}{(1 + e^{-x})^2}

This formula ensures that gradients remain smooth during training.

1. Which of the following correctly represents the derivative of $x^4$ ?

2. The derivative of a constant is always:

3. What is the derivative of $x^5$ ?

4. Given the following sigmoid function:

g(x) = \frac{2}{1+e^{-x}}

What is its derivative $g'(x)$ ?

Which of the following correctly represents the derivative of $x^4$ ?

Select the correct answer

4x^3

x^3

4x^4

3x^2

The derivative of a constant is always:

Select the correct answer

$1$

$C$

Undefined

$0$

What is the derivative of $x^5$ ?

Select the correct answer

x^4

$5x^4$

5x^5

4x^3

Given the following sigmoid function:

g(x) = \frac{2}{1+e^{-x}}

What is its derivative $g'(x)$ ?

Select the correct answer

g(x) + 2

g(x)^2

g(x)(2 - g(x))

2e^{-x}

War alles klar?

Wie können wir es verbessern?

Danke für Ihr Feedback!

Abschnitt 3. Kapitel 3

Introductions to Derivatives

The Limit Definition of a Derivative

Basic Derivative Rules

Power Rule

Constant Rule

Sum & Difference Rule

Product & Quotient Rules

Product Rule

Quotient Rule

Chain Rule: Differentiating Composite Functions

Exponential Chain Rule Example:

Logarithmic Chain Rule Example:

Special Case: Derivative of the Sigmoid Function

1. Which of the following correctly represents the derivative of x4x^4x4?

2. The derivative of a constant is always:

3. What is the derivative of x5x^5x5?

4. Given the following sigmoid function:

Awesome!

Introductions to Derivatives

The Limit Definition of a Derivative

Basic Derivative Rules

Power Rule

Constant Rule

Sum & Difference Rule

Product & Quotient Rules

Product Rule

Quotient Rule

Chain Rule: Differentiating Composite Functions

Exponential Chain Rule Example:

Logarithmic Chain Rule Example:

Special Case: Derivative of the Sigmoid Function

1. Which of the following correctly represents the derivative of x4x^4x4?

2. The derivative of a constant is always:

3. What is the derivative of x5x^5x5?

4. Given the following sigmoid function:

1. Which of the following correctly represents the derivative of $x^4$ ?

3. What is the derivative of $x^5$ ?

1. Which of the following correctly represents the derivative of $x^4$ ?

3. What is the derivative of $x^5$ ?