Exponential and logarithmic functions: further differentiation and integration · Grade 12 · 5 units
In Exponential and logarithmic functions: further differentiation and integration (Grade 12 · 5 units) you practise 6 skills: The derivative of a power with any base · The inverse of an exponential function · The domain of a natural logarithm · The integral of a derivative over its function · The integral of a composite times the inner derivative · Investigating combined functions. The derivative of a power whose base is any positive number comes from the number e: write the base as e to the power of its natural log, and differentiate by the chain rule. So the derivative of such a power is the same power times the natural log of the base. When the base is greater than e the derivative is greater than the function, and when it is less than e the derivative is smaller.
The inverse of an exponential function is a logarithmic function: make the variable the subject with the natural log, then swap the letters. The domain of the inverse is the range of the original function. The natural log is defined only for positive numbers, so the log of an expression is defined only where the expression is positive, and at the end of that interval the graph has a vertical asymptote. When the numerator of a fraction is the derivative of its denominator, the integral of the fraction is the natural log of the denominator, as long as the denominator is positive. In the same way, when a composite function is multiplied by the derivative of its inner expression, the integral is an antiderivative of the outer function, applied to that expression. Always check the answer by differentiating it. To investigate a product of a polynomial and an exponential function, differentiate by the product rule and take out the power of e as a common factor. A power of e is always positive, so the sign of the derivative is the sign of the other factor. When there is a periodic function such as sine inside the power, the function has infinitely many extreme points.
Tip: In the integral of a product or a fraction, look for an expression whose derivative appears as a factor.
What you learn
The derivative of a power with any base
For example: What is the derivative of f(x) = 2ˣ ?
- f′(x) = x × 2^(x − 1)
- f′(x) = 2ˣ
- f′(x) = 2ˣ ÷ ln 2
- f′(x) = 2ˣ × ln 2
Answer and solution
The answer: f′(x) = 2ˣ × ln 2
Here 2ˣ = e^(x ln 2) , so by the chain rule f′(x) = 2ˣ × ln 2 .
The inverse of an exponential function
For example: What is the inverse of the function f(x) = e^(x + 3) + 4 ?
- f⁻¹(x) = ln(x + 4) − 3
- f⁻¹(x) = ln(x + 3) + 4
- f⁻¹(x) = ln(x − 4) + 3
- f⁻¹(x) = ln(x − 4) − 3
Answer and solution
The answer: f⁻¹(x) = ln(x − 4) − 3
From y = e^(x + 3) + 4 we get e^(x + 3) = y − 4 , so x = ln(y − 4) − 3 . Swap the letters: f⁻¹(x) = ln(x − 4) − 3 .
The domain of a natural logarithm
For example: What is the domain of the function f(x) = ln(−2x + 5) ?
Answer and solution
The answer: x < 2.5
The expression −2x + 5 must be positive: −2x + 5 > 0 , so x < 2.5 .
The integral of a derivative over its function
For example: What is the general antiderivative of 2x ÷ (x² + 5) ?
- 1 ÷ (x² + 5) + C
- 2x × ln(x² + 5) + C
- −1 ÷ (x² + 5)² + C
- ln(x² + 5) + C
Answer and solution
The answer: ln(x² + 5) + C
The derivative of the denominator x² + 5 is 2x , which is the numerator. The denominator is always positive, so the antiderivative is ln(x² + 5) + C .
The integral of a composite times the inner derivative
For example: What is ∫ (2x − 3) × e^(x² − 3x) dx ? Hint: the factor 2x − 3 is the derivative of the exponent.
- e^(2x − 3) + C
- (2x − 3) × e^(x² − 3x) + C
- e^(x² − 3x) ÷ (2x − 3) + C
- e^(x² − 3x) + C
Answer and solution
The answer: e^(x² − 3x) + C
By the chain rule, the derivative of e^(x² − 3x) is (2x − 3) × e^(x² − 3x) . So the integral is e^(x² − 3x) + C .
Investigating combined functions
For example: The function f(x) = x × e^(−4x) has one extreme point. What is its x coordinate?
Answer and solution
The answer: 1/4
The derivative is f′(x) = e^(−4x) × (1 − 4x) . It is zero when 1 − 4x = 0 , that is x = 1/4 .
Not on the site yet
Parts of this topic that are not built yet: Powers with a rational exponent (removed from the syllabus) · The volume of a solid of revolution (removed from the syllabus).
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 5 units (grades 10 to 12).