Differentiation
Differentiation is defined as the rate of change of a quantity with respect to another. For example, speed is measured as the rate of change of distance with respect to time. It also helps us to determine the rate of change of variable x with respect to y. The graph of y drawn against x is the gradient of the curve.
This formula list consists of derivatives for constant, trigonometric functions, polynomials, hyperbolic, logarithmic functions, exponential, inverse trigonometric functions, etc.
Differentiation Formulas List
In all the formulas below, f’ means d(f(x))/dx = f′(x) and g’ means d(g(x))/dx = g′(x) . Both f are the functions of x and and g is differentiated with respect to x. We can also represent dy/dx = Dx/y. Some of the general differentiation formulas are;
Power Rule: \[ (d/dx) (x^{n}) = nx(^{n - 1})\]
Derivative of a constant, a: (d/dx) (a) = 0
Derivative of a constant multiplied with function f: (d/dx) (a. f) = af’
Sum Rule: (d/dx) (f ± g) = f’ ± g’
Product Rule: (d/dx) (fg)= fg’ + gf’
Quotient Rule: d/dx(fg) = gf′–fg′g\[_{2}\]
Differentiation Formulas for Trigonometric Functions
d/dx (sin x) = cosx
d/dx (cos x) = –sinx
d/dx (tan x) = sec 2x
d/dx (cot x) = −cosec 2x
d/dx (sec x) = sec x tan x
d/dx (cosec x) =−cosec x cot x
d/dx(sinh x) =cosh x
d/dx(cosh x) =sinh x
d//dx(tanh x) =sec h 2x
d/dx(coth x) =−cosech 2x
d/dx(sech x) =−sech x tanh x
ddx(cosech x) =−cosec h x coth x
Differentiation Formulas for
If y = sin-1 x, y' = \[1 \sqrt{(1-x^{2})}\]
If y = cos-1 x, y' = \[-1 \sqrt{(1-x^{2})}\]
If y = tan-1 x, y' = \[1(1+x^{2})\]
If y = cot-1 x, y' =\[−1(1+x^{2})\]
If y = sec-1 x, y' = \[1x \sqrt{(x^{2}−1)}\]
If y = cosec-1 x, y' = \[−1x\sqrt{(x^{2}−1)}\]
FAQs on Differentiation Formula
1. What is differentiation?
Differentiation is defined as the rate of change of a quantity with respect to another. The speed is measured as the rate of change of distance with respect to time. This speed at each moment is not the same as the calculated average. Speed is similar to the slope, which is nothing but the rate of change of the distance over a period of time.
The ratio of a small change in one quantity causes a small change in another which completely depends on the first quantity. This process is called differentiation. It is one of the important concepts in calculus and it mainly focuses on the differentiation of a function. The differentiation determines the maximum or minimum value of a function, the velocity and acceleration of moving objects, and the tangent of a curve.
If y = f(x) that is differentiable, then the differentiation is denoted as f'(x) or dy/dx.
2. What is Implicit differentiation?
Let f(x,y) be a function with elements x and y. If y cannot be solved directly, implicit differentiation can be used. Suppose f(x,y) = 0, then differentiate this function with respect to x and note the terms containing dy/dx at one side and find dy/dx.
For example, let us find dy/dx if \[x^{2} + y^{2} =1\].
We differentiate both sides of the equation.
\[ \frac{d}{dx.x^{2}} + \frac{d}{dx.y^{2}} = \frac{d}{dx.1}\]
2x + 2y.dy/dx = 0
dy/ dx = -x/y
3. What is the Logarithmic differentiation equation?
If a function is the product and quotient of functions, as in
\[y ={f_{1}(x).f_{2}(x).....}/{g_{1}(x).g_{2}(x).....}\]
we should first take the logarithm and then differentiate it. If a function is an exponent of a function over another, as in \[[f(x)]^{g(x)}\] then the logarithm of the function f(x) should be taken (to base e) and then differentiate it.
For example, if \[y = x^{x}\] , then log y = x log x
1/y. dy/dx = log x + 1
dy/dx = y. (log x + 1)
= \[ x^{x}\](log x + 1)
4. Derive higher-order differentiation?
We can find the higher-order derivatives on successive differentiation. The further differentiation of the first derivative is denoted by f'' or \[\frac{d^{2}y}{dx^{2}}\] and the third derivative is denoted by f'" or \[d^{3}y dx^{3}\]. The nth derivative of f(x) is defined as f n(x) is used in the power series.
For example, the rate of change of displacement is referred to as the velocity. The second derivative of displacement is referred to as the acceleration and the third derivative is referred to as the jerk.
Let us consider a function, y = f(x) = \[x^{5} - 3x^{4} + x\]
f1(x) = \[5x^{4} - 12x^{3} + 1\]
f2(x) = \[20x^{3} - 36x^{2}\]
f3(x) = \[60x^{3} - 76x\]
f4(x) = 120x -72
5. What are some of the important notes for differentiation?
Differentiation of a function is to find the rate of change of the function with respect to another quantity. f′(x)= [limΔx→0 f(x+Δx)−f(x)] / Δx, where Δx is the incremental change in x.
Differentiation is the process of finding the derivatives of the function if the limit exists. The derivative of a function is dy/dx or y' or f'(x).
Differentiability depends on continuity, but its converse is not true.
6. When Should We Use the Chain Rule and the Product Rule in Differentiation?
Both of the two rules are very useful for differentiating functions. The chain rule is used when we differentiate a 'function of a function', like f(g(x)) in general while the product rule is used when we are calculating the derivative of the two functions that are being multiplied together, like f(x)g(x) in general.
For example, f(x) = sin(3x) is the basic example of what we call a 'composite' function or a 'function of a function'. In this example, the two functions are as follows: function one takes x and then multiplies it by 3 while function two takes the sine of the answer given by function one. While differentiating equations we use the chain rule for these types of functions.
On the contrary, if the function in question would have been, say, f(x) = xcos(x), then we know that it is time to use the product rule. This is because there are two separate functions multiplied together: 'x' takes x but does nothing; 'cos(x)' takes the cosine of x. Note that they're separate functions: one that doesn't rely on the answer to the other!
7. What Does the Quotient Rule Mean?
The quotient rule comes to play its role when a function is divided by another function and you need to find the derivative of it. It is one of the last main rules that we use to calculate derivatives.
We can begin with f(x) = \[ x / x^{2}\]. What can be the derivative of f(x)? Is it just the top or maybe the bottom? will it be the derivative of the top times or the derivative of the bottom? What might it be? Well, we can always simplify this into 1 / x, and then f`(x) would then be \[ -1 / x^{2}\] because 1 / x is the same as \[x^{-1}\]. Then we use the power rule. But is there a way that we can find without simplifying it? No, we won't be able to simplify it in every case. Think about sin(x) / x, or \[(x^{2} + ln(x)) / cos(x)\]. What about those cases? Perhaps, you must know the quotient rule.