Lesson 6.2: Differentiation Review and Standard Derivatives

Introduction

Welcome to Lesson 6.2 in Foundation Further Mathematics! In this lesson, we will explore differentiation, a fundamental concept in calculus that helps us understand how functions behave. You will learn about the product, quotient, and chain rules, as well as the derivatives of standard functions like exponential, logarithmic, and trigonometric functions. Our primary objectives are:

• Apply the product, quotient, and chain rules effectively.
• Differentiate exponential, logarithmic, and trigonometric functions.
• Understand and use higher derivatives and notation.
• Differentiate products, quotients, and composite functions with fluency.
• Quote and use the derivatives of all standard functions.

Are you ready to dive into the world of differentiation? Let’s get started! 🚀

Differentiation Basics

Differentiation is the process of finding the rate at which a function changes. Simply put, it helps us find the slope of the tangent line to a curve at any given point. The derivative of a function $f(x)$ is often denoted as $f'(x)$ or $\frac{df}{dx}$.

The Power Rule

One of the most basic and widely used rules in differentiation is the power rule. The power rule states:

$$\frac{d}{dx}(x^n) = n x^{n-1}$$

This means if you have a function like $f(x) = x^3$, its derivative would be:

$$f'(x) = 3x^{3-1} = 3x^2$$

Constant Rule

The derivative of a constant is always zero:

$$\frac{d}{dx}(c) = 0$$

Where $c$ is a constant. For example, the derivative of $f(x) = 5$ is:

$$f'(x) = 0$$

Sum Rule

The sum rule states that the derivative of a sum of functions is the sum of their derivatives:

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

For example, if $f(x) = x^2$ and $g(x) = 3x$, then:

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

The Product Rule

When dealing with the product of two functions, we use the product rule:

$$\frac{d}{dx}(u \cdot v) = u'v + uv'$$

Let’s say $u = x^2$ and $v = \sin(x)$, then first, we find $u'$ and $v'$:

• $u' = 2x$
• $v' = \cos(x)$

Using the product rule, we differentiate:

$$\frac{d}{dx}(x^2 \cdot \sin(x)) = (2x) \cdot \sin(x) + (x^2) \cdot (\cos(x))$$

The Quotient Rule

For functions divided by each other, we use the quotient rule:

$$\frac{d}{dx}\left(\frac{u}{v}

ight) = $\frac{u'v - uv'}{v^2}$$$

For instance, if $u = x^2$ and $v = \cos(x)$, we find:

• $u' = 2x$
• $v' = -\sin(x)$

Thus, applying the quotient rule gives us:

$$\frac{d}{dx}\left(\frac{x^2}{\cos(x)}

ight) = $\frac{(2x)(\cos(x)) - (x^2)(-\sin(x))}{\cos^2(x)}$$$

The Chain Rule

The chain rule is essential for differentiating composite functions. It states:

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

If we have $f(x) = \sqrt{x}$ and $g(x) = 3x + 1$, we find:

1. $f'(x) = \frac{1}{2\sqrt{x}}$
2. $g'(x) = 3$

Thus, using the chain rule:

$$\frac{d}{dx}(\sqrt{3x + 1}) = \frac{1}{2\sqrt{3x + 1}} \cdot 3$$

Standard Derivatives

Here are some common derivatives you should memorize:

• $$\frac{d}{dx}(e^x) = e^x$$
• $$\frac{d}{dx}(\ln(x)) = \frac{1}{x}$$
• $$\frac{d}{dx}(\sin(x)) = \cos(x)$$
• $$\frac{d}{dx}(\cos(x)) = -\sin(x)$$
• $$\frac{d}{dx}( an(x)) = \sec^2(x)$$

These derivatives are the building blocks for more complex calculus problems.

Higher Derivatives

The nth derivative is simply the derivative of the derivative. It allows us to analyze the behavior of functions further. For example, if:

• $f'(x)$ is the first derivative,
• $f''(x)$ is the second derivative, and so on.

If you have $f(x) = x^3$, then:

• $$f'(x) = 3x^2$$
• $$f''(x) = 6x$$
• $$f'''(x) = 6$$

Conclusion

In this lesson, we have reviewed the key concepts of differentiation, covering the fundamental rules and derivatives of standard functions. Understanding these concepts is vital as we move forward in calculus, especially when we explore integration and more advanced topics. Keep practicing these rules to become fluent in differentiation! ✌️

Study Notes

• The Power Rule: $$\frac{d}{dx}(x^n) = n x^{n-1}$$
• The Constant Rule: $$\frac{d}{dx}(c) = 0$$
• The Sum Rule: $$\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$$
• The Product Rule: $$\frac{d}{dx}(u \cdot v) = u'v + uv'$$
• The Quotient Rule: $$\frac{d}{dx}\left(\frac{u}{v}

ight) = $\frac{u'v - uv'}{v^2}$$$

• The Chain Rule: $$\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$$
• Memorize standard derivatives: $$\frac{d}{dx}(e^x), \frac{d}{dx}(\ln(x)), \frac{d}{dx}(\sin(x)), \frac{d}{dx}(\cos(x)), \frac{d}{dx}( an(x))$$
• Higher derivatives represent deeper analysis into function behavior.