Derivatives of basic elementary functions - Differentiation Rules

Derivatives of basic elementary functions

We shall now find the derivatives of all the basic elementary functions; we start with the constant function.

(1) The derivative of a constant function is zero.

Let y = f(x) = k, k is a constant.

Then f(x +∆x) = k and

f(x + ∆x) - f(x) = k - k = 0.

(2) The power function y = xn, n > 0 is an integer.

Let f(x) = xn.

Then, f(x+ ∆x) = (x + ∆x)n and

f(x + ∆x) - f(x) = (x + ∆x)n - xn.

(3) Derivative of the logarithmic function

The natural logarithm of x is denoted by loge x or logx or ln x

Let y =  f (x) = log x

Now  f (x + ∆x) = log (x + ∆x) and

f(x + ∆x) - f(x) = log (x + ∆x) - log x

(4) Derivative of the exponential function

Let y = ax

Then f (x + ∆x) - f (x) = ax +Δx− ax

= ax (a ∆x−1) and

(5) The derivatives of the Trigonometric functions

(i) The sine function, sinx.

Let y = f (x ) = sin x .

Then f (x + ∆x) = sin(x + ∆x) and

(ii) The cosine function, cos x

(iii) The tangent function, tan x

Let y = f (x) = tan x .

(iv) The secant function,  sec x

(v) The cosecant function, cosec x

(vi) The cotangent function, cot x

(6) The derivatives of the inverse trigonometric functions

(i) The derivative of arcsin x or sin−1 x

Let y= f (x) = sin−1 x .

Then  y + ∆y= f (x + ∆x ) = sin−1 (x +  ∆x)

This implies, x= sin y and

x + ∆x= sin (y + ∆y).

(ii) The derivative of arccos x or cos−1x

We know the identity :

(iii) The derivative of arctan x or tan−1 x

Let y = f (x ) = tan−1 x .            ... (1)

This implies, y + ∆y= f(x + ∆x) = tan−1 (x + ∆x)          ... (2)

x= tan y and

x+ ∆x = tan (y + ∆y)

This implies, ∆x = tan (y + ∆y) - tan y

(iv) The derivative of arccot x or cot−1 x

(v) The derivative of arcsec x or

(vi) The derivative of arccosec x or

The proofs of (v) and (vi) are left as exercises.

EXERCISE 10.2

Find the derivatives of the following functions with respect to corresponding independent variables:

(1) f(x) = x - 3 sinx

(2) y = sin x + cos x

(3) f(x) = x sin x

(4) y = cos x - 2 tan x

(5) g(t) = t3cos t

(6) g(t) = 4 sec t + tan t

(7) y = ex sin x

(8) y = tanx /x

(9) y = sin x / [1+ cos x]

(10) y = x / [sin x + cos x]

(11) y = tan x −1 / sec x

(12) y = sin x / x2

(13) y = tan θ(sin θ + cos θ)

(14) y = cosec x . cot x

(15) y = x sin x cos x

(16) y = e-x. log x

(17) y = (x2 +5) log(1+x)e-3x

(18) y = sin x°

(19) y = log10  x

(20) Draw the function f '(x) if f (x) = 2x2 − 5x + 3