Differentiating exponential and logarithmic functions · Grade 12 · 4 units
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
For example: Between which two whole numbers is the number e ?
- 1 < e < 2
- 3 < e < 4
- 2 < e < 3
Answer and solution
The answer: 2 < e < 3
Its value is about e ≈ 2.718 , so 2 < e < 3 .
The derivative of an exponential function
For example: Here is a function: f(x) = −4 × e^(4x) Which of these is f′(x) ?
- −4 × e^(4x)
- 4 × e^(4x)
- −16 × e^(4x − 1)
- −16 × e^(4x)
Answer and solution
The answer: −16 × e^(4x)
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
For example: Here is a function: f(x) = ln(x + 6) Which of these is f′(x) ?
- x ÷ (x + 6)
- 1 ÷ x
- 1 ÷ (x + 6)²
- 1 ÷ (x + 6)
Answer and solution
The answer: 1 ÷ (x + 6)
The function inside the logarithm is x + 6 , and its derivative is 1 . So the derivative is 1 ÷ (x + 6)
Products and quotients
For example: Here is a function: f(x) = x² × ln x Which rule do you use first to differentiate it?
- The quotient rule
- The chain rule
- The sum rule
- The product rule
Answer and solution
The answer: The product rule
The function is a product of two factors, so use the product rule.
The equation of a tangent
For 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 and solution
The answer: 4x − 4
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
For example: Here is the function f(x) = a × eˣ − 5 , and f(0) = 0 . What is a ?
Answer and solution
The answer: 5
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
Parts of this topic that are not built yet: The second derivative.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).