Differentiation Laws

Derivatives are the fundamental tool of calculus. The derivative of a real variable function measures the sensitivity to change the function value when its argument is changed. 

Calculus Differentiation Rules

The derivative laws are the standard rules used for computing the derivative of a function in calculus.

Here, let us look into all the elementary derivative rules which are applied in calculus when finding the derivative of any function.

1) Constant Rule

According to this rule, the differentiation of any constant value is 0.

If f(x) = c where c is any constant

Then the derivate is f'(x) = 0.

2) Differentiation of a Function is Linear.

This rule provides a derivative of any function when performing addition and subtraction on the functions. 

For any function f(x) and g(x) with any real numbers p and q. The derivative of this function is linear.

h(x) = a f(x) + b g(x)

The differentiation of this function is linear.

h'(x) = a f'(x) + b g'(x)

The special cases of linearity rule are:

• Constant Multiple Rule

(a f(x))' = a f'(x)

• Addition Rule

(f+g)' = f' + g'

• Subtraction Rule

(f-g)' = f' - g'

3) Product Rule

This rule provides us with the derivative of functions when they are multiplied with each other.

For any function f(x) and g(x) the derivative of the function h(x) = f(x) g(x) with respect to x by product rule is 

h'(x) = f'(x) g(x) + f(x) g'(x)

4) Chain Rule

This rule provides us with the derivative of a composite function.

For any function f(x) and g(x) the derivative of the function h(x) = f(g(x)) with respect to chain rule is

h'(x) = f'(g(x)) . g'(x)

5) Polynomial or Elementary Power Rule

The combination of the power rule with the sum and constant multiple rules allows the calculation of any polynomial's derivatives.

If f(x) = xr where r is any real number not equal to zero. Then the derivative is

f'(x) = r xr-1

When r = 1 then the function becomes a special case f(x) = x, so the derivative is f'(x)=1.

6) Quotient Rule

The quotient rule is used to find the derivative of a function when the ratio of two differentiable functions is given.

If f(x) and g(x) are two functions such that 

h(x) = f(x)/ g(x)

The derivative of a function by quotient rule is 

h'(x) = f'(x) g(x) + f(x) g'(x) / (g(x))2

7) The Derivative of Trigonometric Functions 

The differentiation of trigonometric functions is the mathematical process for identifying the derivative or its rate of change in relation to a variable of a trigonometric function.

If f(x) = sin (x) then f'(x) = cos (x)

If f(x) = cos (x) then f'(x) = -sin (x)

If f(x) = tan (x) then f'(x) = sec2 (x)

If f(x) = sec (x) then f'(x) = sec (x) tan (x)

If f(x) = cot (x) then f'(x) = -cosec2 (x)

If f(x) = -cosec (x) then f'(x) = -cosec (x) cot (x)

8) The Derivative of an Exponential Function 

An exponential function is a Mathematical function of form f (x) = ax, where x is a variable and a is a constant which is called the base of the function which is greater than 0. The differentiation of exponential function is the mathematical process of finding the rate of change in relation to a variable.

If f(x) = ax then f'(x) = ln (a) ax

If f(x) = ex then f'(x) = ex

If f(x) = ag (x) then f'(x) = ln (a) ag (x) g'(x)

If f(x) = eg (x) then f'(x) = eg (x) g'(x)

9) The Derivative of Logarithmic Functions

The inverse of exponential functions is Logarithmic functions. Logarithmic differentiation is a process used to simplify certain terms by using logarithms and their differentiation rules before effectively applying derivatives. Exponent removal, product conversion into sums and division into subtraction, which can lead to a simplified expression to derivatives, can be utilised with logarithm.

If f(x) = loga (x) then f'(x) = 1 / ln (a) x

If f(x) = ln (x) then f'(x) = 1/ x

If f(x) = loga (g(x)) then f'(x) = g'(x) / ln (a) g(x)

If f(x) = ln (g(x)) then f'(x) = g'(x) / g(x)

Problems on Calculus Differentiation Rules

1) Find the Derivative of Under Root x.

Ans: Here the given function is f(x) = \[\sqrt{x}\] 

The function can be written as f(x) = (x)\[^{1/2}\]

Now differentiating the function by applying elementary power derivative rule we get 

f'(x) = ½ (x)\[^{-1/2}\] 

f\[^{1}\](x) = \[\frac{1}{2\sqrt{x}}\]

Therefore, the derivative of under root x is \[\frac{1}{2\sqrt{x}}\].

2) Find the Differentiation of x Cube.

Ans: Here the given function is f(x) = x3

To find the differentiation of x cube we will apply the elementary power derivative rule

f'(x) = 3x2 is the derivative of the function x3.

3) Find the Derivative of e to the Power x.

Ans: Here the given function is f(x) = ex

We will use exponential function differentiation laws to find the derivative of e power x.

So f'(x) = ex is the derivative of the function ex.

4) Find x Power x Derivative.

Ans: Here the given function is f(x) = xx

By making use of exponential function we can write f(x) = e\[^{ln(x^{x})}\] where x = e\[^{ln x}\]

Now we will use logarithmic functions differentiation laws to find the x power x derivative.

So f'(x) = (1 + ln x) xx is the derivative of the function xx.

5) Find the Derivative of e.

Ans: Here, the given function is f(x) = e

Where e is a constant function that is equal to [1+(1/n)]n. So by using the constant rule of differentiation we get 

f'(x) = 0 is the derivative of the function e.  

Conclusion

• A derivative is the rate of change in the variable of a function.

• The derivative helps us to the slope of a function at any point.

• Derivatives are crucial to solving calculus and differential equation problems.