Derivative of Polynomials and Trigonometric Functions

In mathematics, polynomial functions are the functions that involve only non-negative integer powers, i.e. only positive integer exponents of a variable such as 3x2 + 5, 2x3 – 7x – 5, and so on. When we extend the definition of trigonometric ratios to any angle in terms of radian measure then we treat them as trigonometric functions and they are sin x, cos x, tan x, cosec x, sec x and cot x. In this article, you will understand how to find the derivatives of different types of polynomial functions and trigonometric functions along with formulas and examples.

Derivative of Polynomial Functions

Let \(f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+…+a_{1}x+a_{0}\) be a polynomial function, then the derivative function is given by:

\(\frac{d}{dx}f(x)=\frac{d}{dx}[a_{n}x^{n}+a_{n-1}x^{n-1}+…+a_{1}x+a_{0}]\)

This can be expanded as:

\(\frac{df(x)}{dx}=na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+…+2a_{2}x+a_{1}\)

Here, a1, a2,…, an are all real numbers and an ≠ 0.

Derivatives of Polynomial Formulas

To find the derivative of a given polynomial function, it is required to get thoroughly familiar with the following basic derivatives formulas and rules. These are used while calculating the derivative of a simple or complex polynomial function.

  • \(\large \frac{d}{dx}(c)=0\)
  • \(\large \frac{d}{dx}(x)=1\)
  • \(\large \frac{d}{dx}(x^{n})=nx^{n-1}\)
  • \(\large \frac{d}{dx}(u\pm v)=\frac{du}{dx}\pm \frac{dv}{dx}\)
  • \(\large \frac{d}{dx}(cu)=c\frac{du}{dx}\)
  • \(\large \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)
  • \(\large \frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}\)

We know that some functions involve trigonometric functions also along with the variables. In that case, we need to find the derivative of trigonometric functions along with the other terms. Hence, one requires the knowledge of how to find the derivatives of these functions. Let’s have a look at the process of finding the derivative of a function in trigonometry along with formulas.

Derivative of Trigonometric Functions

Finding the derivative of a trigonometric function involves the use of limits. Suppose the derivative of sin x can be calculated as given below:

Derivative of sin x:

Let f(x) = sin x

We know that,

\(\frac{df(x)}{dx}=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\)

Here, f(x) = sin x and f(x + h) = sin(x + h)

Now,

\(\frac{df(x)}{dx}=\lim_{h\rightarrow 0}\frac{sin(x+h)-sin\ x}{h}\)

Using the formula sin C – sin D = 2 cos[(C + D)/2] sin[(C – D)/2],

\(\frac{df(x)}{dx}=\lim_{h\rightarrow 0}\frac{2cos\frac{(2x+h)}{2}sin(\frac{h}{2})}{h}\\=\lim_{h\rightarrow 0}\frac{cos\frac{(2x+h)}{2}sin(\frac{h}{2})}{\frac{h}{2}}\)

By applying product rule of limits,

\(\frac{df(x)}{dx}=\lim_{h\rightarrow 0}cos(x+\frac{h}{2}).\lim_{h\rightarrow 0}\frac{sin\ \frac{h}{2}}{\frac{h}{2}}\)

As we know \(\lim_{x\rightarrow 0}\frac{sin\ x}{x} = 1\),

\(\frac{df(x)}{dx}=cos\ x . 1 = cos\ x\)

Therefore, the derivative of sin x is cos x, i.e. (d/dx)(sin x) = cos x.

Derivatives of Trigonometric Functions Formulas

  • \(\large \frac{d}{dx}(\sin\ x)=\ cos\ x\)
  • \(\large \frac{d}{dx}(\cos\ x)=-\sin\ x\)
  • \(\large \frac{d}{dx}(\tan\ x)=\sec^{2}(x)\)
  • \(\large \frac{d}{dx}(\cot\ x)=-\csc^{2}(x)\)
  • \(\large \frac{d}{dx}(\sec\ x)=\sec\ x\ tan\ x\)
  • \(\large \frac{d}{dx}(cosec\ x)=-cosec\ x\ cot\ x\)

Click here to get more differentiation formulas.

Solved Examples

Example 1:

Find the derivative of the polynomial function f(x) = x100 + x99 + x98 + …. + x2 + x + 1 at x = 1.

Solution:

Given,

f(x) = x100 + x99 + x98 + …. + x2 + x + 1

Using the formulas d/dx (xn) = nxn-1 and (d/dx)k = 0,

(d/dx)f(x) = (d/dx) [x100 + x99 + x98 + …. + x2 + x + 1]

= (d/dx)x100 + (d/dx)x99 + (d/dx)x98 + …. + (d/dx)x2 + (d/dx)x + (d/dx)1

= 100x99 + 99x98 + 98x97 + …. + 2x + 1 + 0

= 1 + 2x + …. + 98x97 + 99x98 + 100x99

Now, the derivative of f(x) at x = 1 is,

1 + 2(1) + …. + 98(1)97 + 99(1)98 + 100(1)99

= 1 + 2 + …. + 98 + 99 + 100

= [100(100 + 1)]/2 {sum of first 100 natural numbers}

= 50 × 101

= 5050

Example 2:

Find the derivative of f(x) = x sin x – 4x2.

Solution:

Given

f(x) = x sin x – 4x2

(d/dx) f(x) = (d/dx) [x sin x – 4x2]

= (d/dx) (x sin x) – (d/dx)4x2

Using the product and power rule of differentiation,

= x [(d/dx) sin x] + sin x [(d/dx) x] – 4 (d/dx)x2

= x cos x + sin x (1) – 4 (2x)

= x cos x + sin x – 8x

Practice Problems

1. Find the derivative of the function 7 tan x – 2 sec x.

2. Find the derivative of f(x) = 2x – (x/4).

3. Find the derivative of x2 – 2 at x = 10.

4. Compute the derivative of f(x) = sin2x.

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