## 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* + Δ*y* = sin (*v +* Δ*v*)*.*

*Step* 2. Δ*y* = sin (*v +* Δ*v*) − 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.**

Let*y =* 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 =* sin^{2} *x.*

*Solution. y* = sin^{2} *x* = (sin *x*)^{2}.

Here *v* = sin *x,* and *n* = 2.

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

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* sin^{3} *x*.

*Solution.*

Here *u* = cos *x*, and *v* = sin^{3} *x*.

= cos *x*(3 sin^{2} *x* cos *x*) + sin^{3} *x*(− sin *x*)

= sin^{2} *x*(3 cos^{2} *x* − sin^{2} *x*).

**EXERCISE 5—5**

*Differentiate:*

**1.** *y* = sin 3*x*

**2.** *y* = cos 5*ax*

**3.** *y* = sin^{2} *x* cos *x*

**4.** *ρ* = *a* cos 2*θ*

**5.** *y* = sin (*a* + *bx*)

**6.** *y* = e^{tan x}

**7.** ρ = log cos^{2} *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 2*θ*