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

Tuesday, May 23, 2017

Derivative of a composite function


Derivation of a composite function

Let's consider a function g. The image by g of any element x of its domain is g(x). Let's consider another function f. The image of g(x) by f is f[g(x)] also written as fog(x). The function fog is called the composed function of g and f.

If g is differentiable for any element x and f is differentiable at g(x) fog(x) = f[g(x)] is differentiable at x. The derivative of the function fog is (fog)'(x) = f''[g(x)].g'(x), The demonstration of this formula is not done here.

The derivative of fog is the product of the derivative of fog by the derivative of g.

If u is a function of x and f is a function of u then f(u) is a composite function. By applying the rule above the derivative of f(u) or f'(u) is equal to the derivative of f(u) multiplied by the derivative of u. We write [f(u)]' = f''(u).u'. If we introduce the notation (d) of differentiability we can write d/dx[f(u)] =d/du[f(u)].du/dx.

In practical applications we have a function f to differentiate with respect to x. We then introduce a function u that is a function of x. Now we have the composite function f(u). The diferentiation or derivative of f with respect to x is equal to the derivative of f with respect to u multiplied by the derivative of u with respect to x . This derivative is called the chain rule. There is a chain of operations to do. First we introduce a new function u. Then we calculate the derivative of f as the the composite function f(u) by applying the formula for the derivative of a composite function.

The chain rule holds also the application of the power rule when we work with a complex function with exponents.

The power rule applies by introducing the new function u.

Example 1 

Let's calculate the derivative of the function f(x) = (2x+1)²

To make the computation of the derivative easy we introduce the function u. Then the function f becomes f(x) = u². The derivative of the function f with respect to x is the derivative of the expression u² with respect to x . We write d/dx[f(x)] = d/dx[u²]

By applying the formula for the derivative of a composed function we have d/dx[f(x)] = d/du(u²).du/dx.

By calculating d/du(u²) we obtain d/dx[f(x)] = 2u. u'

Let's substitute u: d/dx[f(x)] = 2 (2x+1)(2x+1)'

By calculating the derivative of 2x+1 we obtain d/dx[f(x)] = 2(2x+1)(2) = 4(2x+1) = 8x+1

Example 2

Calulate the derivative of f(x) = (x²+3x+4)²

Let's write u = x^2+3x+4

d/dx[f(x)] = d/dx(x²+3x+4)²
                =  d/dx(u²)
               =   d/du(u²).du/dx (Applying the formula of the derivative of a composite function)
               =   2u.u'
               =   2(x²+3x+4)(x²+3x+4)' (Substituting u)
               =   2(x²+3x+4)(2x+3)
              =    2(2x³+6x²+8x+3x²+9x+12)
              =    2(2x³+9x²+17x+12)
              =     4x³+18x²+34x+24)

Interested in learning more about the techniques of calculations for derivatives visit this site and subscribe to the Calculus course





Friday, May 12, 2017

Derivative computations

The formula lim f(x)-f(x+h)/h when  x→h that defines the derivative of a function f implies tedious calculations to calculate the derivative of some types of functions and combinations of functions..

Therefore some formulas have been established to determine the derivatives of a combination of functions and some specific types of functions.

The formulas for the constant function and the power functions are called respectively constant rule and power rule. The formulas for the sum, product and quotient of functions are called respectively addition rule, product rule and quotient rule. The derivative of a composition of 2 functions f and g is called the chain rule. It is an extension of the power rule The trigonometric, logarithmic and exponential functions have their specific formula.

The derivative of an implicit function is called implicit differentiation.

It is essential to memorize the formulas. Otherwise, it would be difficult to calculate the derivatives of these particular functions. Today we are going to limiting ourselves to the learning of the basic formulas: constant, power, sum, product and quotient rule.

Derivative of a constant

The derivative of the function constant is 0. If f(x) = c the derivative of f(x) is 0. We write:  f′(x) = 0.


The Power rule

The derivative of the function power defined by f(x) =  xn is equal to n multiplied by x to the power of n-1. The formula is .f’(x) = nxn-1

Derivative of the product of a constant by a function

The derivative of the product of a function by a constant is equal to the product of the constant by the derivative of the function power.

If  f(x) = axn its derivative is f’(x) = axn-1


Derivative of the function f(x) = x

The derivative of the function f(x) = x can be calculated using the formula for the derivative of the function power. In order to use this formula we have to write f(x) = x as the function power. We write f(x) = x as f(x) = x
By applying the formula for the function power we obtain f’(x) = x1-1 
 f’(x) = x0 ⇒ f’(x) = 1 

Derivative of a sum of functions

If f. g. h,;;; are differentiable for any value of x of their domain the derivative of the sum of these functions is f’+g’+h’+ ....

Derivative of the product of 2 functions

If f and g are differentiable for any value x of their domain the derivative of the product f.g is fg’+gf’

Derivative of the quotient of 2 functions

If f and g are differentiable for any value of their domain the derivative of the quotient f/g is (f∕g)’ = f’g-gf’∕g2

These formulas have to be demonstrated and the learners have to do some exercises to apply them. If anyone is interested in learning more subscribe to these courses via this link Free Introductory Calculus Course and Complete Calculus Course


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