Saturday, May 27, 2017

Derivative of Trigonometric functions

In this post I will state the formulas for the derivative of trigonometric functions. I will also give some techniques to remember them and solve problems. I'll do some examples and give some exercises for practice. The formulas will not be demonstrated here.

It's not sufficient to know the formulas for the derivative of trigonometric functions to be able to calculate the derivative of functions containing trigonometric expressions. The calculations of these functions involve being able to apply all the other rules that enable to calculate the derivative of a function.

Derivative of the function sine

The derivative of the function sine is equal to the function cosine. If f(x) = sinx f''(x) or d/dx(sinx) = cosx

Derivative of the function cosine

The derivative of the function cosine is equal to the opposite of the function sine. If f(x) = cosx f'(x) or d/dx(cosx) = -sinx

Derivative of the function tangent

The derivative of the function tangent is equal to the square of the secant function. If f(x) = tanx f'(x) or d/dx(tanx) = sec²x

Derivative of the function cotangent

The derivative of the function cotangent is equal to the opposite of the square of the cosecant function. If f(x) = cotx  f''(x) or d/dx(cotx) = -csc²x

Derivative of the function secant

The derivative of the function secant is equal to the product of the function secant by the function tangent. If f(x) = secx f'(x) or d/dx(secx) = secx.tanx

Derivative of the function cosecant

The derivative of the function cosecant is equal to the opposite of the product of the function cosecant by the function cotangent. If f(x) = cosecx f'(x) or d/dx(cosecx) = -cosecx.cotx.

Observations that allow to memorize the formulas

1) All the derivatives of co-functions have the negative sign. For examples, the derivative of cosx = -sinx, the derivative of cotx = -cosec²x, the derivative of cosecx = -cosecx.tanx
2) For the sine and cosine functions the derivative of the first function is equal to the second function The derivative of sinx is cosx. The derivative of the second function is equal to the opposite of the first function. The derivative of cosx is -sinx
3) When thinking about the drivative of the tangent and cotangent functions think about the the square of the function secant and cosecant. The derivative of the tangent goes with the square of the secant Example the derivative of tanx = sec²x. The derivative of cotangent goes with the square of the cosecant preceded by the negative sign, Example the derivative of cotx = -csc²x
4) For the derivative of the functions secant and cosecant think about multiplying the function secant by the function tangent and the cosecant by cotangent. Example the derivative of secx = secx.tanx. The derivative of coscx = -coscx.cotx In the case of the derivative of the cosecant don't forget to place negative placed before the product.  

Example 1

If f(x) = x²cosx+sinx find f'(x)

The derivative of a sum of two functions is equal to the sum of the derivatives of each function.
f'(x) = (x²cosx)'+(sinx)'
 Applying the product rule to calculate the derivative of x²cosx
f'(x) = (x²)'(cosx) + (x²) (cosx)'+ cosx. I apply the formula (uv)' = u'v+uv'
f'(x) =  2xcosx + (x²)(-sinx) + cosx
        =  2xcosx-x² sinx+cosx
        = -x²sinx + 2xcosx + cosx.

Example 2

If f(x) = sin²x find f'(x)
Let's write f(x) as f(x) = (sinx)²
Let's write sinx = u. Then f(x) = u² and f(u) = u²
The function f becomes the function composite f(u)
The derivative of the composite function f(u) is f'(u) = f'(u).u'
Since f(x) and f(u) are both equivalent we have f(x) = f'(u).u'
f'(x) = 2u,u'
       = 2sinx.(sinx)' (Substituting u)
       = 2sinxcosx.


Example 3

Find the derivative of f(x) = sinx-1/sinx+1

Applying the quotient rule f'(x) = (sinx-1)'(sinx+1)-(sinx-1)(sinx+1)'/(sinx+1)²
Calculating the derivatives: f'(x) = cosx(sinx+1)-(sinx-1)cosx/(sinx-1)²
f'(x) = sinxcosx+cosx-sinxcosx+cosx/(sinx-1)²
f'(x) - cosx/(sinx-1)²

Practice

1) What are the techniques to memorize the formulas of the derivative of the following functions
a) sine and cosine
b) tangent and cotangent
c) secant and cosecant

2) Calculate the derivatives of the following functions:
a) f(x) - xsinx+4
b) f(x) = xcox-x²tanx-2
c) f(x) = cos³x
d) f(x) = cosx+sinx/cosx-sinx

Interested in knowing more about derivatives visit this site Center for Integral Development