Product rule, Quotient rule, Tri functions Derivatives

Main Operation in Derivative Problems

Identify the primary operation before taking a derivative: addition/subtraction, multiplication (Product Rule), division (Quotient Rule), or composite functions (Chain Rule).

Power Rule Formula

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

Derivative of a Constant Formula

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

Differentiating $a x^n$ Formula

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

Condition for Product Rule

When two functions of $x$ are multiplied together.

Product Rule Formula

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

Steps for Product Rule

1. Identify $f(x)$ and $g(x)$.
2. Find $f'(x)$ and $g'(x)$.
3. Substitute into $f'(x) g(x) + f(x) g'(x)$.

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

$$2x \sin(x) + x^2 \cos(x)$$

Common Mistake when Differentiating $x^2 \sin(x)$

Incorrectly taking the derivative of both parts individually as $2x \cos(x)$ without applying the Product Rule.

Memory Pattern for Product Rule

Think: D-L + L-D (Differentiate first, Leave second + Leave first, Differentiate second).

$$\frac{d}{dx}(2x \sin(x))$$

$$2 \sin(x) + 2x \cos(x)$$

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

$$2x \cos(x) - x^2 \sin(x)$$

When NOT to Use Product Rule

Do not use the Product Rule when multiplying a function by a constant scalar, such as $4 x^3$ (use the Power Rule instead).

Condition for Quotient Rule

When one function of $x$ is divided by another function of $x$.

Quotient Rule Formula

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

Steps for Quotient Rule

1. Find $f'(x)$ and $g'(x)$.
2. Substitute into the formula: $\frac{g(x) f'(x) - f(x) g'(x)}{(g(x))^2}$

$$\frac{d}{dx}\left(\frac{x^2}{x+1}\right)$$

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

Common Mistakes in Quotient Rule

Forgetting the subtraction sign in the numerator, failing to square the denominator, or reversing the order of terms in the numerator.

$$\frac{d}{dx}(\sin(x))$$

$$\cos(x)$$

$$\frac{d}{dx}(\cos(x))$$

$$-\sin(x)$$

$$\frac{d}{dx}(\tan(x))$$

$$\sec^2(x)$$

$$\frac{d}{dx}(\cot(x))$$

$$-\csc^2(x)$$

$$\frac{d}{dx}(\sec(x))$$

$$\sec(x) \tan(x)$$

$$\frac{d}{dx}(\csc(x))$$

$$-\csc(x) \cot(x)$$

Trigonometric Derivatives with Negative Signs

The derivatives of co-functions: $\cos(x)$, $\cot(x)$, and $\csc(x)$.

Two Fundamental Trigonometric Limits

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ and $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$

$$\lim_{x \to 0} \frac{x}{\sin(x)}$$

$$1$$

$$\lim_{x \to 0} \frac{\tan(x)}{x}$$

$$1$$

$$\lim_{x \to 0} \frac{x}{\tan(x)}$$

$$1$$

$$\lim_{x \to 0} \frac{1 - \cos(x)}{x^2}$$

$$\frac{1}{2}$$

Meaning of $f'(x)$, $f''(x)$, $f'''(x)$, and $f^{(4)}(x)$

They denote the 1st, 2nd, 3rd, and 4th derivatives of $f(x)$, respectively.

First derivative of $f(x) = x^4 - 2x^3 + x^2 - 8x + 17$

$$f'(x) = 4x^3 - 6x^2 + 2x - 8$$

Second derivative of $f(x) = x^4 - 2x^3 + x^2 - 8x + 17$

$$f''(x) = 12x^2 - 12x + 2$$

Third derivative of $f(x) = x^4 - 2x^3 + x^2 - 8x + 17$

$$f'''(x) = 24x - 12$$

Fourth derivative of $f(x) = x^4 - 2x^3 + x^2 - 8x + 17$

$$f^{(4)}(x) = 24$$

First derivative of $f(x) = 2x \sin(x) + x^2 \cos(x) + e^x + 1$

$$f'(x) = 2 \sin(x) + 4x \cos(x) - x^2 \sin(x) + e^x$$

Second derivative of $f(x) = 2x \sin(x) + x^2 \cos(x) + e^x + 1$

$$f''(x) = 6 \cos(x) - 6x \sin(x) - x^2 \cos(x) + e^x$$

Third and Fourth derivatives of $f(x) = 2x \sin(x) + x^2 \cos(x) + e^x + 1$

Third derivative: $f'''(x) = -12 \sin(x) - 8x \cos(x) + x^2 \sin(x) + e^x$ Fourth derivative: $f^{(4)}(x) = -20 \cos(x) + 10x \sin(x) + x^2 \cos(x) + e^x$

Memory Patterns for Derivative Rules

Power Rule: Bring exponent down, subtract 1. Product Rule: D-L + L-D. Quotient Rule: Low d-high minus high d-low over low squared. Trig Negatives: Derivatives of COS, COT, CSC are negative.