8. Derivative Rules and Derivatives of Basic Functions

Differentiation rules allow us to break down complex functions into manageable parts. The most error-prone case is the composite function—both the outer and inner layers’ rates of change must be accounted for.

I begin by identifying the functional hierarchy. When encountering a composite function, I first write the derivative of the outer function, then multiply it by the derivative of the inner function.

In the previous section, we explored the definition and geometric interpretation of derivatives, gaining insight into how derivatives quantify the instantaneous rate of change of a function. Next, we delve into differentiation rules and the derivatives of fundamental functions—laying essential groundwork for upcoming applied examples.

I. Differentiation Rules

Differentiation rules are a set of systematic techniques that enable efficient computation of derivatives. Mastering these rules significantly streamlines differentiation, especially for complicated functions. Below are some commonly used rules:

When learning differentiation rules, first identify which elementary components constitute the function, then decide whether to apply the sum/difference, product, quotient, or chain rule.

1. Constant Rule

If $c$ is a constant, its derivative is zero:

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

2. Power Rule

For any real number $n$:

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

3. Sum Rule

If $f(x)$ and $g(x)$ are differentiable functions, then the derivative of their sum is:

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

4. Product Rule (Leibniz Rule)

For differentiable functions $f(x)$ and $g(x)$, the derivative of their product is:

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

5. Quotient Rule

For differentiable functions $f(x)$ and $g(x)$, where $g(x) \neq 0$, the derivative of their quotient is:

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

6. Chain Rule

For a composite function $y = f(g(x))$, its derivative is:

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

II. Derivatives of Basic Functions

Having learned the core differentiation rules, we now turn to the derivatives of key elementary functions—knowledge essential for rapid and accurate differentiation in practice.

You need not absorb every detail of “Differentiation Rules and Derivatives of Basic Functions” all at once. Start with a small, hands-on problem you can verify yourself, then use the diagrams and text to fill in conceptual gaps.

1. Exponential Functions

The derivative of the natural exponential function $e^x$ is:

$$\frac{d}{dx}(e^x) = e^x$$

For a general exponential function $a^x$, where $a > 0$:

$$\frac{d}{dx}(a^x) = a^x \ln(a)$$

2. Logarithmic Functions

The derivative of the natural logarithm is:

$$\frac{d}{dx}(\ln x) = \frac{1}{x}, \quad x > 0$$

For the logarithm base $a$:

$$\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}, \quad x > 0$$

3. Trigonometric Functions

Derivatives of basic trigonometric functions:

• $\displaystyle \frac{d}{dx}(\sin x) = \cos x$
• $\displaystyle \frac{d}{dx}(\cos x) = -\sin x$
• $\displaystyle \frac{d}{dx}(\tan x) = \sec^2 x$

4. Inverse Trigonometric Functions

• $\displaystyle \frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 - x^2}}, \quad -1 < x < 1$
• $\displaystyle \frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1 - x^2}}, \quad -1 < x < 1$
• $\displaystyle \frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2}$

III. Example Application

Let’s apply these rules and derivatives to a concrete example.

Consider the function:

$$f(x) = 3x^3 - 5x^2 + 2x - 7$$

We compute its derivative using the sum rule and power rule:

1. Differentiate each term individually:

   • Derivative of $3x^3$ is $9x^2$
   • Derivative of $-5x^2$ is $-10x$
   • Derivative of $2x$ is $2$
   • Derivative of the constant $-7$ is $0$

2. Combine results:

$$f'(x) = 9x^2 - 10x + 2$$

We can verify this result programmatically using Python:

import sympy as sp

x = sp.symbols('x')
f = 3*x**3 - 5*x**2 + 2*x - 7
f_prime = sp.diff(f, x)
f_prime

Running this code yields the derivative $f'(x) = 9x^2 - 10x + 2$.

After reading “Differentiation Rules and Derivatives of Basic Functions”, organize your understanding into a recap table: first clarify the central thread, then test it with a small task to confirm correctness.

Upon finishing “Differentiation Rules and Derivatives of Basic Functions”, select a simple example and walk through the full differentiation process end-to-end. Then assess which steps you can now execute independently.

IV. Summary

In this section, we covered foundational differentiation rules and the derivatives of essential elementary functions. These tools prepare us thoroughly for the next section, where we’ll explore practical applications—including tangent lines and real-world rate-of-change problems. Equipped with these rules and derivatives, we’re ready to tackle increasingly sophisticated challenges. Next, we’ll examine how derivatives manifest in concrete contexts.