The Calculus Primer (2011)

Part V. Differentiation of Transcendental Functions

Chapter 18. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS

5—9.The Derivative of the Sine. To derive the formulas for the derivatives of the six trigonometric functions, we need only fall back upon the General Rule to find the derivative of the sine; the derivatives of the other five functions may be derived from this one by trigonometric transformations. It should be recalled (§1—18) that

Now let

y = sin v,

where v is any given function of the angle x expressed in radians.

Step 1. y + Δy = sin (v + Δv).

Step 2. Δy = sin (v + Δv) − sin v.(2)

By a trigonometric transformation, the difference between two sines is given by

sin A − sin B = 2 cos (A + B) sin (A − B);

applying this to equation (2), we have:

By equation (1) above,

Step 4. = cos v.

But v is a function of x; hence to find , we use the relation

Substituting the value of from Step 4 in equation (3), we obtain:

5—10.Derivative of the Cosine.

Lety = cos v.

From trigonometry:

Differentiating equation (1) by formula [6]:

But by trigonometry.

Hence equation (2) becomes:

5—11.Derivative of the Tangent and the Cotangent.

Differentiating equation (1) as a quotient:

Again, let

Differentiating equation (1) as a quotient:

5—12.Derivative of the Secant and the Cosecant. These may be derived in a similar manner, and are left as an exercise for the reader. The formulas obtained are as follows:

5—13.Differentiating Trigonometric Functions. The formulas for differentiating trigonometric functions may be used in conjunction with the formulas for differentiating algebraic exponential and logarithmic functions, as suggested by the following examples.

EXAMPLE 1.Differentiate y = sin² x.

Solution. y = sin² x = (sin x)².

Here v = sin x, and n = 2.

Then = 2(sin x) (cos x) = sin 2x.

EXAMPLE 2.Differentiate y = e^x·sin x.

Solution.

Here u = e^x, and v = sin x.

= e^x·cos x + sin x·e^x

= e^x (sin x + cos x).

EXAMPLE 3.Differentiate y = (sin x)^{sin x}.

Solution.

Here u = sin x, and v = sin x.

Using §5—7, formula [5]:

= sin x·sin x^{(sin x−l)}·cos x + log sin x·(sin x)^sinx·(cos x),

or = cos x (sin x)^{sin x}·(1 + log sin x).

EXAMPLE 4.Differentiate y = cos x sin³ x.

Solution.

Here u = cos x, and v = sin³ x.

= cos x(3 sin² x cos x) + sin³ x(− sin x)

= sin² x(3 cos² x − sin² x).

EXERCISE 5—5

Differentiate:

1. y = sin 3x

2. y = cos 5ax

3. y = sin² x cos x

4. ρ = a cos 2θ

5. y = sin (a + bx)

6. y = e^{tan x}

7. ρ = log cos² θ

8. y = (sin x)^x

9. y = x^{cos x}

10. y = e^x·log sin x

11. ρ = log tan θ

12. y = (cos x)^x

17. ρ = θ − sin θ cos θ

18. y = 2 sin x cos x

19. y = log x + tan x cos x

20. ρ = log sin 2θ