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Topos Uranos
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Derivatives of the elementary functions

Summary

This lesson differentiates the elementary functions that the preceding lesson did not reach: sine and cosine, from the trigonometric limits of unit 1; tangent, cotangent, secant and cosecant, with the quotient and reciprocal rules; the natural exponential and the natural logarithm, from their standard limits; and the hyperbolic functions, from the exponential. No proof uses the chain rule or the inverse function rule, which come later: in each case, a sum formula or a law of the exponential or of the logarithm splits the difference quotient into a factor that depends only on the point and another that is an already known limit. With these derivatives and the rules, the lesson computes tangents and angles between transcendental curves, and fulfills what the preceding lesson announced: the function x2sin⁡1xx^2 \sin \frac{1}{x}, differentiable at 00, has a derivative without a limit at 00.

Learning objectives

  1. Prove that the derivative of sine is cosine and the derivative of cosine is the negative of sine, and explain why the variable is measured in radians.
  2. Differentiate tangent, cotangent, secant and cosecant with the rules of the preceding lesson, and specify the domain of each derivative.
  3. Prove that the natural exponential is its own derivative and that the derivative of ln⁡∣x∣\ln \abs{x} is 1x\frac{1}{x}, and obtain from this the derivatives of the hyperbolic functions.
  4. Combine the elementary derivatives with the rules to find tangents and angles between curves, and recognize that a derivative may have no limit at a point where the function is differentiable.

Sine and cosine

The lesson on differentiation rules differentiated all the functions obtained from the identity and the constants by the four operations: polynomials and rational functions. Sine is not one of them, and its derivative must come from its difference quotient. At 00 that quotient is sin⁡hh\frac{\sin h}{h}, and the fundamental trigonometric limit, from the lesson on the squeeze theorem, gives the derivative 11. For an arbitrary point aa, the formula for the sine of a sum splits sin⁡(a+h)\sin(a + h) into a part that depends only on aa and another that depends only on hh; the quotients that remain are two limits from the corollary on deduced trigonometric limits, in the same lesson: sin⁡hh\frac{\sin h}{h} tends to 11, and 1−cos⁡hh\frac{1 - \cos h}{h} tends to 00.

TheoremDerivatives of sine and cosine

Sine and cosine are differentiable on all of R\mathbb{R}, and for every real xx:

ddx(sin⁡x)=cos⁡x,ddx(cos⁡x)=−sin⁡x\frac{\mathrm{d}}{\mathrm{d}x}\left(\sin x\right) = \cos x, \qquad \frac{\mathrm{d}}{\mathrm{d}x}\left(\cos x\right) = -\sin x

The proof uses the increment form of the derivative, from the lesson on the derivative as a limit, and the sum formulas from the lesson on the sum and difference formulas. The figure follows the tangent line to the graph of sine: its slope at each point is the ordinate of cosine at the same point; at the end, the tangent line to the graph of cosine has as its slope the negative of sine.

Proof

  1. sin⁡(a+h)−sin⁡a=sin⁡acos⁡h+cos⁡asin⁡h−sin⁡a=sin⁡a(cos⁡h−1)+cos⁡asin⁡h\sin(a + h) - \sin a = \sin a \cos h + \cos a \sin h - \sin a = \dato{s}{\sin a \left(\cos h - 1\right) + \cos a \sin h}

    Let aa be a real number and h≠0h \neq 0. By the formula for the sine of a sum, the increment of sine splits into two terms: one with the factor cos⁡h−1\cos h - 1 and another with the factor sin⁡h\sin h; the factors sin⁡a\sin a and cos⁡a\cos a do not depend on hh.

  2. sin⁡(a+h)−sin⁡ah=−sin⁡a⋅1−cos⁡hh+cos⁡a⋅sin⁡hh→−sin⁡a⋅0+cos⁡a⋅1=cos⁡a\frac{\sin(a + h) - \sin a}{h} = -\sin a \cdot \frac{1 - \cos h}{h} + \cos a \cdot \frac{\sin h}{h} \to -\sin a \cdot 0 + \cos a \cdot 1 = \resaltar{\cos a}

    Divide by hh. By the corollary on deduced trigonometric limits, the two quotients tend to 00 and to 11; by the theorem on the limit of a linear combination, from the algebra of limits, the difference quotient has a limit, and the increment form gives the derivative of sine at aa.

  3. cos⁡(a+h)−cos⁡ah=−cos⁡a⋅1−cos⁡hh−sin⁡a⋅sin⁡hh→−cos⁡a⋅0−sin⁡a⋅1=−sin⁡a\frac{\cos(a + h) - \cos a}{h} = -\cos a \cdot \frac{1 - \cos h}{h} - \sin a \cdot \frac{\sin h}{h} \to -\cos a \cdot 0 - \sin a \cdot 1 = \resaltar{-\sin a}

    With the formula for the cosine of a sum, cos⁡(a+h)=cos⁡acos⁡h−sin⁡asin⁡h\cos(a + h) = \cos a \cos h - \sin a \sin h, the same reasoning gives the derivative of cosine.

The proof uses the fact that the variable is measured in radians: the fundamental limit compares the sine with the length of the arc. If the angle were measured in degrees, the sine of xx degrees would be sin⁡πx180\sin \frac{\pi x}{180}, and its derivative would carry the factor π180\frac{\pi}{180} (proposed problem 1). Radians are, therefore, the unit in which sine has cosine as its derivative, with no factors. The geometric reading of the theorem is that of the figure: sine increases where cosine is positive and has a horizontal tangent where cosine vanishes; the tangent line to sine at the origin is the line y=xy = x, and that is why, near 00, sin⁡x\sin x resembles xx, as the equivalence sin⁡x∼x\sin x \sim x from the lesson on equivalent infinitesimals already said.