AP® Calculus AB · Unit 2 · Topic 2.7
Derivatives of cos x, sin x, ex, and ln x
Four functions have derivatives worth memorizing: sin x gives cos x, cos x gives -sin x, ex gives ex, and ln x gives 1/x. Each comes from a limit, not a shortcut rule.
Look at the sine curve and ask where it is rising fastest, flat, and falling fastest. At x = 0, sine is climbing at its steepest, and cosine happens to equal 1 there. At x = π/2, sine flattens at its peak, and cosine equals 0 there. That matching pattern, all the way around, is why the slope graph of sine traces out cosine exactly. Cosine's own slope graph works the same way but flipped, giving -sin x.
The algebra behind this comes from the limit definition of the derivative applied to sin(x). Expanding sin(x+h) with the angle addition formula splits the difference quotient into two pieces, each built from one of two special limits: sin(h)/h approaches 1 as h approaches 0, and (cos(h)-1)/h approaches 0. Those two facts alone force the derivative of sin x to collapse down to cos x. The derivative of ex comes from a similar limit, (eh - 1)/h approaching 1, which only happens because e is defined as the base that makes that limit true. The derivative of ln x follows from treating it as the inverse of ex and using implicit differentiation.
These four derivatives become building blocks the moment a function mixes polynomials with trig, exponential, or log pieces. Differentiating term by term, each piece keeps its own rule: power rule for xn, and these special results for sin x, cos x, ex, ln x. They also set up everything coming next in the unit, since the product, quotient, and chain rules all get practiced on combinations of these exact functions.
Key terms
- difference quotient
- The expression (f(x+h) - f(x))/h whose limit as h approaches 0 defines the derivative.
- special limit
- A specific limit, like sin(h)/h as h approaches 0 equaling 1, used to derive a known derivative formula.
- e
- The unique constant, about 2.71828, for which ex is its own derivative.
- inverse function
- A function that reverses another; ln x is the inverse of ex, which lets its derivative be found from ex's.
Watch it explained
- How it worksHow does the special limit of sin(h)/h produce the derivative of sine?
- Worked exampleFind the derivative of f(x) = 3 sin(x) + 5 ln(x) at x = 1.
- Common mistakeWhy isn't the derivative of ex found by dropping the exponent down in front?
- Going furtherWhat makes e the one base whose exponential is its own derivative?
Check yourself
End of topic.