--- 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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 The derivative of e^x is e^x. For a positive constant base a, (a^x)' = a^x ln a. Taking a = e gives e^x ln e = e^x because ln e = 1. For a differentiable inner function g, the chain rule gives (e^(g(x)))' = e^(g(x))g'(x). Keep the exponent unchanged and multiply by its derivative. *(drawing: The chain rule multiplies the exponential factor by the derivative of its exponent.)* A real logarithm needs a positive argument. For x > 0, (ln x)' = 1/x. Since 1/x is positive there, the logarithm has an upward slope. For differentiable g with g(x) > 0, (ln(g(x)))' = g'(x)/g(x). For x > 0, ln(7x) = ln 7 + ln x, so its derivative is 1/x. For ln(9-3x), first require 9-3x > 0, giving x < 3. Use radians. For 0 < |h| < pi/2, unit-circle area bounds give cos h <= sin h/h <= 1. Also |cos h-1| <= h²/2. Squeezing gives sin h/h → 1 and (cos h-1)/h → 0. The proof reading derives these bounds without derivatives. Thus (sin x)' = cos x and (cos x)' = -sin x. For differentiable g, keep g(x) and multiply by g'(x); cos(5x) gives -5 sin(5x). Evaluate the derivative for a tangent slope; zero means horizontal. **Example.** For x > 0, differentiate x^4 ln x using the product rule (uv)' = u'v + uv'. The power rule gives (x^4)' = 4x^3, and the log rule gives (ln x)' = 1/x. Add the two terms: 4x^3 ln x + x^4(1/x). Since x^4/x = x^3, the derivative is 4x^3 ln x + x^3. For a > 0, write a^x = e^(x ln a). The exponent has derivative ln a, giving (a^x)' = a^x ln a. Thus gradient divided by height equals ln a. If the gradient is m and the height is v, solve ln a = m/v. For base e this ratio is 1. Among positive bases, only e makes the exponential's gradient equal its height everywhere. Your small-angle work used radians. At zero, sin x has value 0 and slope cos 0 = 1. Its tangent is y = x, so sin x ≈ x near zero. For example, the tangent estimates sin(0.07) as 0.07. This is an estimate, not an exact identity. Since the derivative of cos x is -sin x, its gradient near zero is about -x. **Recap.** Keep inner arguments and multiply by their derivatives. Logarithms need positive arguments. These trig rules use radians, with sin x ≈ x near zero. *Lesson, questions and flashcards written for this learning dataset. Mathematical reference records were consulted to check facts and methods.* ## Practice 16 questions on this page, each with its working shown. ## Needs first - [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) ## Optional connections - [The exponential function and the number e](https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ) ## 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)