Theorems of Derivatives

The algebraic process for calculating derivatives is known as differentiation. The slope or gradient of a particular graph at any given position is the derivative of a function. The tangent’s value drawn to that curve at any given location is the gradient of that curve. The curve’s gradient varies at different positions along the axis for non-linear curves. As a result, calculating the gradient in such situations is challenging.

It is also known as a property’s change about another property’s unit change.

Consider the function f(x) as a function of the independent variable x. The independent variable x is caused by a tiny change in the independent variable Δx. The function f(x) undergoes a similar modification Δf(x). 

The ratio is: Δf(x)Δx, and is a measure of f(x) of change in relation to x.

As Δx approaches zero, the ratio’s limit value is limΔx0f(x)Δx  and is known as the first derivative of the function f(x).

Theorems of Derivatives

Theorem 1: Differentiability Implies Continuity

• f is differentiable, meaning  f’(C) exists, then f is continuous at c.

Theorem 2: The Constant Rule

• The derivative of a constant function is 0.

So, if f(x)=k, f'(x)=0.

Theorem 3: The Power Rule 

The power of a variable, n, can also be rational or fractional, so the variable could have exponents, which are real numbers. 

ddxxn = n xn-1.

Then, to find the derivative,

1. Move the exponent down in front of the variable.
2. Multiply it by the coefficient.
3. Decrease the exponent by 1.

Theorem 4: The First Principle Rule

The first principle is “The derivative of a function at a value is the limit at that value of the first part or second derivative”. This principle defines the limit process for finding the derivative at a certain value because all functions have limits.

Theorem 5: The Sum and Difference Rules

If f(x) = u(x) ± v(x), then;

f ‘(x) = u'(x) ± v'(x).

Theorem 6: Derivatives of Trigonometric Functions

• When we differentiate sin x, we get cos x.
• When we differentiate cos x, we get -sin x.
• When we differentiate tan x, we get sec2 x.
• When we differentiate cot x, we get -cosec2 x.
• When we differentiate sec x, we get secx tanx.
• When we differentiate cosec x, we get -cosec x cot x.

Theorems 7: The Product Rule

The product rule of calculus, also known as the Leibniz rule, is used to find the derivative of any given function present in a product form of two differentiable functions. 

The product rule is used to find the derivative of a function in the form of f(x).g(x), in which both the f(x) and g(x) are differentiable entities. 

ddx[f(x)g(x)]= f'(x)g(x)+f(x)g'(x).

Theorem 8: The Quotient Rule

The quotient rule is a calculus method for obtaining the derivatives of any function given as a quotient obtained by dividing two differentiable functions. According to the quotient rule, the ratio of the outcome is similar to the derivative of a quotient formed by subtracting the numerator multiplied by the denominator’s derivatives from the denominator multiplied by the denominator’s derivatives to the denominator’s square.

ddxf(x)g(x)=f'(x)g(x)-f(x)g'(x)[g(x)]2.

Theorem 9: The Chain Rule

The chain rule is usually the sole technique to distinguish a composite function. We won’t be able to differentiate correctly if we don’t realize that a function is composite and that the chain rule must be applied. 

 z(x)=f(x)g(x)

z(x)=f(x)g-1(x)

z'(x)=f'(x)g-1(x)+f(x)ddxg-1(x)

z'(x)=f'(x)g-1(x)+f(x)(-1)(g-2(x))g'(x)

z'(x)=f'(x)g(x)–f(x)g'(x)g(x)2

z'(x)=f'(x)g(x)-f(x)g'(x)g(x)2=ddxf(x)g(x).

Numericals on Theorems of Derivatives 

1. z(x)=x5-cosxsinx

After using the quotient rule,

z'(x)=x5-cosx‘sinx-x5-cosxsinx‘sinx2

z'(x)=5x4-sinxsinx-x5-cosxcosxsinx2

z'(x)=1+5x4sinx-x5cosxsinx2.

1. z(x)=x+cosxtanx

After using the quotient rule,

z'(x)=x+cosx‘tanx-x+cosxtanx‘tanx2

z'(x)=1-sinxtanx-x+cosxsec2xtanx2.

Conclusion

In general terms, differentiation in mathematics is the process of finding the derivative of any function. In scientific terms, it is a rate of change in some of the functions. The basic step of finding the derivative of a function is taking a limit of difference. However, it becomes tedious to repeat every step. There are various rules for differentiation that will enable finding the derivative to mitigate this. The article discussed the fundamental theorems and formulas for the differentiation above.