Derivatives of Exponential, Logarithmic and Trigonometric Functions
Establish the derivatives of the standard non-polynomial functions, including why e is the base that makes the exponential its own derivative.
What a learner can do afterwards
- Differentiate e^(3x), ln(5x) and cos(2x)
- Say what is special about the base e in terms of gradients
- Use the derivative of sin x to explain the small-angle approximation
1 · Read
Three rows cover this topic. The derivative of e^x is e^x, of a^x is a^x ln a, and of ln x is 1/x. For trig, sin x gives cos x and cos x gives -sin x. With the chain rule: e^(3x) gives 3e^(3x), ln(5x) gives 1/x since ln 5 is constant, and cos(2x) gives -2 sin(2x).
Take x squared ln x and name the rules as you go. The product rule opens it: 2x ln x + x squared times 1/x. The power rule gave 2x, the log rule gave 1/x, and the last term simplifies to x. So the answer is 2x ln x + x.
The base e is special because ln e = 1, so the extra ln factor vanishes and e^x reproduces itself. Other bases keep it: 2^x gives 2^x ln 2. Since the slope of sin x at 0 is cos 0 = 1, small angles obey sin x near x, so sin(0.1) is about 0.1.
Pair each function with its row, let the chain rule add the inner factor, and watch constants vanish.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.