# Differentiation Rules

Mathematics I · Differential Calculus · https://tryals.app/learn/mathematics-i/differentiation-rules

## Basic & Combination Rules

Evaluating limits every time is impractical. **Differentiation rules** make derivative calculation routine algebra.

| Rule | Formula | Example |
| --- | --- | --- |
| **Power rule** | $\frac{d}{dx}x^n = nx^{n-1}$ | $\frac{d}{dx}x^3 = 3x^2$ |
| **Constant multiple** | $\frac{d}{dx}(cf) = cf'$ | Constants factor out |
| **Sum rule** | $(f+g)' = f' + g'$ | Sum of derivatives |

For **combining functions**, use these rules:

* **Product rule**: $(fg)' = f'g + fg'$. Differentiate one factor at a time and add.
* **Quotient rule**: $(f/g)' = \frac{f'g - fg'}{g^2}$. Note the minus sign and order.
* **Chain rule**: $(f(g(x)))' = f'(g(x))g'(x)$. Outer times inner.

## Standard Toolkit & Pitfalls

Combine the combination rules with standard derivatives:

| Function | Derivative | Function | Derivative |
| --- | --- | --- | --- |
| $\sin x$ | $\cos x$ | $e^x$ | $e^x$ |
| $\cos x$ | $-\sin x$ | $\ln x$ | $1/x$ |

**Common Pitfalls:**

* **Product rule error**: The derivative of a product is **not** the product of derivatives. $(fg)' \neq f'g'$. Always use $f'g + fg'$.
* **Chain rule error**: For $f(g(x))$, you **must** multiply by $g'(x)$. For example, $\frac{d}{dx}\sin(x^2) = \cos(x^2)\cdot 2x$, **not** just $\cos(x^2)$. Forgetting $g'(x)$ is the #1 mistake.

## Practice questions

5 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. The derivative of a sum is the sum of the derivatives, but the derivative of a product is not the product of the derivatives. Why does one operation pass straight through differentiation and the other not?

A. Because products of functions are not themselves functions, so the derivative must be defined separately.
B. Because the product rule was derived later, and a simpler rule may yet be found.
C. Because the sum rule is an approximation that happens to be exact, while the product rule is exact by construction.
D. Because differentiation is linear and multiplication is not.

**Answer:** D. Because differentiation is linear and multiplication is not.

**Why:** The counterexample is immediate: f = g = x gives (x²)' = 2x while the product of the derivatives is 1. Picture a rectangle with sides f and g, growing both adds a strip along the top and one along the side, f'g and fg', plus a corner square that vanishes in the limit. That corner is why the naive guess fails and the answer has two terms.

Page: https://tryals.app/practice/mathematics-i/differentiation-rules/the-derivative-of-a-sum-is-the-sum-of-the-derivatives-but-the

### 2. The chain rule is the rule that makes almost every other derivative usable in practice. What is it actually saying about composed change?

A. That composed functions change more slowly than their components
B. That the inner function must be differentiated first for the order to work out
C. That rates multiply through a chain: three times two is six
D. That a composition can be differentiated by differentiating each part and adding the results

**Answer:** C. That rates multiply through a chain: three times two is six

**Why:** Rates compose by multiplying, which the gear train makes obvious and the notation dy/du · du/dx makes memorable. It is also why backpropagation works: a neural network is a long composition, and the sensitivity of the loss to an early weight is a product of local derivatives along the chain.

Page: https://tryals.app/practice/mathematics-i/differentiation-rules/the-chain-rule-is-the-rule-that-makes-almost-every-other-derivative

### 3. Select every derivative below that is CORRECT.

A. the derivative of (x squared)(x cubed) is 2x times 3 x squared
B. the derivative of ln x is 1/x
C. the derivative of sin(x squared) is cos(x squared)
D. the derivative of cos x is -sin x
E. the derivative of x to the seventh is 7 x to the sixth

**Answer:** B. the derivative of ln x is 1/x; D. the derivative of cos x is -sin x; E. the derivative of x to the seventh is 7 x to the sixth

**Why:** Option 2 drops the inner derivative 2x. Option 4 multiplies the derivatives: the product is x to the fifth, with derivative 5 x to the fourth, whereas the false rule gives 6 x cubed. The other three are standard results.

Page: https://tryals.app/practice/mathematics-i/differentiation-rules/select-every-derivative-below-that-is-correct

### 4. Order the steps of differentiating $y = \sin(3x^{2} + 1)$ by the chain rule.

**Answer:**

1. Identify the outer function sine and the inner function 3 x squared + 1
2. Differentiate the outer function, leaving the inner alone, to get cos(3 x squared + 1)
3. Differentiate the inner function to get 6x
4. Multiply the two results to get 6x cos(3 x squared + 1)

**Why:** Step 1 is the step people skip, and skipping it is why step 3 gets forgotten. The final answer keeps the inner expression intact inside the cosine and multiplies by its derivative outside, omit that factor and the answer is wrong by exactly 6x.

Page: https://tryals.app/practice/mathematics-i/differentiation-rules/order-the-steps-of-differentiating-y-sin-3x-1-by-the-chain-rule

### 5. Sort each claimed derivative as correct or incorrect.

**Answer:**

- Correct: d/dx of x to the fifth is 5x to the fourth, The exponential function is its own derivative
- Incorrect: d/dx of sin(x squared) is cos(x squared), d/dx of (x times sin x) is cos x

**Why:** Items 2 and 5 both drop an inner derivative, of 2 in each case. Item 4 is correct: the quotient rule gives ((x + 1) - x) over (x + 1) squared = 1/(x + 1) squared. Item 6 is correct too, since (1/x squared) times 2x = 2/x, the chain rule was applied and simplified.

Page: https://tryals.app/practice/mathematics-i/differentiation-rules/sort-each-claimed-derivative-as-correct-or-incorrect