# Differentiating exponential and logarithmic functions · Grade 12 · 4 units

> Explanation, worked examples and free practice following the Israeli curriculum.

In Differentiating exponential and logarithmic functions (Grade 12 · 4 units) you practise 6 skills: The number e · The derivative of an exponential function · The derivative of a logarithmic function · Products and quotients · The equation of a tangent · Finding a parameter.

The number e is a special number, about 2.718. The exponential function whose base is e is its own derivative: the derivative of e to the power of x is e to the power of x. The logarithm whose base is e is called the natural logarithm.

The derivative of the natural log of x is 1 over x. When the power, or the inside of the logarithm, is a function, multiply by its derivative: the derivative of e to the power of 3x is 3 times e to the power of 3x, and the derivative of the natural log of x squared plus 1 is 2x over x squared plus 1.

Products, quotients and composite functions follow the same rules as for polynomials. The derivative of the product x times e to the power of x is e to the power of x plus x times e to the power of x. The derivative of the quotient the natural log of x over x is 1 minus the natural log of x, all over x squared. The derivative of the composite function e to the power of x squared is 2x times e to the power of x squared.

The equation of a tangent is found as for polynomials: the gradient is the value of the derivative at the point of contact. Points where the index of e is 0, or where the inside of the logarithm is 1, keep the working simple: e to the power of 0 is 1, and the natural log of 1 is 0.

**Tip:** The value of e to the power of 0 is 1, and the natural log of 1 is 0. At points like these the working is easy.

## What you learn

### The number e

- Example: Between which two whole numbers is the number e ?
- Options: 1 < e < 2 · 3 < e < 4 · 2 < e < 3
- Answer: 2 < e < 3
- Solution: Its value is about e ≈ 2.718 , so 2 < e < 3 .

### The derivative of an exponential function

- Example: Here is a function: f(x) = −4 × e^(4x) Which of these is f′(x) ?
- Options: −4 × e^(4x) · 4 × e^(4x) · −16 × e^(4x − 1) · −16 × e^(4x)
- Answer: −16 × e^(4x)
- Solution: The derivative of the power is 4 , so multiply the function by it: (−4) × 4 × e^(4x) , that is f′(x) = −16 × e^(4x)

### The derivative of a logarithmic function

- Example: Here is a function: f(x) = ln(x + 6) Which of these is f′(x) ?
- Options: x ÷ (x + 6) · 1 ÷ x · 1 ÷ (x + 6)² · 1 ÷ (x + 6)
- Answer: 1 ÷ (x + 6)
- Solution: The function inside the logarithm is x + 6 , and its derivative is 1 . So the derivative is 1 ÷ (x + 6)

### Products and quotients

- Example: Here is a function: f(x) = x² × ln x Which rule do you use first to differentiate it?
- Options: The quotient rule · The chain rule · The sum rule · The product rule
- Answer: The product rule
- Solution: The function is a product of two factors, so use the product rule.

### The equation of a tangent

- Example: Here is a function f: 4 ln x What is the equation of the tangent to its graph at the point where x = 1 ?
- Answer: 4x − 4
- Solution: At the point of contact, f(1) = 4 ln 1 = 0 . The derivative is 4 ÷ x , so the gradient is f′(1) = 4 . The tangent is y = 4x − 4

### Finding a parameter

- Example: Here is the function f(x) = a × eˣ − 5 , and f(0) = 0 . What is a ?
- Answer: 5
- Solution: At x = 0 the value of e⁰ is 1, so a − 5 = 0 and a = 5 . Check: 5 × 1 − 5 = 0

## Not on the site yet

- The second derivative

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 12 · 4 units topics](https://kixplo.com/il/en/math/grade-12-4u/)