---
title: "Derivatives of Exponential, Logarithmic and Trigonometric Functions"
description: "Establish the derivatives of the standard non-polynomial functions, including why e is the base that makes the exponential its own derivative."
canonical: https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT
source: https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT.md
retrieved: 2026-09-12
---

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# 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.

Subject: Mathematics · Area: Calculus & Analysis · Ages 18 to 19
Page: https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT

## Ready when they can

- 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

## Lesson: Derivatives of e, logs and trig

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).

**Example.** 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.

**Recap.** Pair each function with its row, let the chain rule add the inner factor, and watch constants vanish.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [The Exponential Function and the Number e](https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ)
- [Small Angle Approximations](https://lightmysky.com/learn/mathematics/small-angle-approximations-mt_fIsZes5Os3)
- [The Chain Rule](https://lightmysky.com/learn/mathematics/the-chain-rule-mt_kdhl4dmwJn)

## Opens up

- [Implicit Differentiation and Derivatives of Inverse Functions](https://lightmysky.com/learn/mathematics/implicit-differentiation-and-derivatives-of-inverse-functions-mt_P4ABBGsKFT)
- [Maximum Likelihood Estimation](https://lightmysky.com/learn/mathematics/maximum-likelihood-estimation-mt_ZRTrcy7-KV)