Question

Show that following functions are differentiable for every value of x :

(i) Identity function, f (x) = x

(ii) Constant function,f (x) = c, where c is a constant

(iii) f(x) = e^x

(iv) f(x) = sin x.

Answer 1

(i) Given, f (x) = x, (identity function)

where, x ∈ R

Let a be arbitrary constant, then

At x = a, Left hand derivative of f (x)

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So, for every x, identity function f(x) is differentiable.

(ii) Given, constant function f(x) = c, where c is constant. Domain of function f(x) is set of real numbers (R).

Let a be any arbitrary real number, then

At x = a, Left hand derivative of f (x)

So, for every x, identity function f(x) is differentiable.

(iii) Given function f (x) = e^x, where x ∈ R

Let a be an arbitrary constant then at x = a,

Left hand derivative of f (x)

Again, at x = a, Right hand derivative of f (x)

Hence,f(x) = e^x is differentiable for every x.

(iv) Given function f(x) = sin x, where x ∈ R

Let a be any arbitrary real number.

At x = a, Left hand derivative of (x)

Again, at x = a, Right hand derivative of f (x)

Hence, for every x, function will be differentiable.