Derivatives

A derivative in calculus is the rate of change of a quantity y with respect to another quantity x. It is also termed the
differential coefficient of y with respect to x. Differentiation is the process of finding the derivative of a function.

Meaning of Derivatives in Calculus

The derivative of a function f(x) is usually represented by d/dx (f(x)) (or) df/dx (or) Df(x) (or) f'(x). Let us see what a
derivative technically means. Consider a curve of function f(x) and let two points on it be (x, f(x)) and ((x + h), f(x +
h)). Then the slope of the secant line through these points is given by [f(x + h) - f(x)]/(x + h - x) = [f(x + h) - f(x) / h.
See the figure below and observe that when the distance between two points is closely equal to 0 (i.e., as h
approaches 0), the second point overlaps the original point and the secant line becomes the tangent line. In
calculus, the slope of the tangent line is referred to as the derivative of the function. i.e.,

The derivative of the function, f '(x) = Slope of the tangent = limh→0 [f(x + h) - f(x) / h.

This formula is popularly known as the "limit definition of the derivative" (or) "derivative by using the first
principle".

Interpretation of Derivatives

The derivative of a function f(x) in math is denoted by f'(x) and can be contextually interpreted as follows:

The derivative of a function at a point is the slope of the tangent drawn to that curve at that point.

It also represents the instantaneous rate of change at a point on the function.

The velocity of a particle is found by finding the derivative of the displacement function.

The derivatives are used to optimize (maximize/minimize) a function.

Thus, whenever we see the phrases like "slope/gradient", "rate of change", "velocity (given the displacement)",
"maximize/minimize" etc then it means that the concept of derivatives is involved.

Derivative of a Function Using the First Principle

The derivative of a function can be obtained by the limit definition of derivative which is f'(x) = limh→0 [f(x + h) -
f(x) / h. This process is known as the differentiation by the first principle. Let f(x) = x2 and we will find its derivative
using the above derivative formula. Here, f(x + h) = (x + h)2 as we have f(x) = x2. Then the derivative of f(x) is,

f '(x) = limh→0 [(x + h)2 - x2] / h

= limh→0 [ x2 + 2xh + h2 - x2] / h

= limh→0 [ 2xh + h2] / h

= limh→0 [ h(2x + h) ] / h

= limh→0 (2x + h)

= 2x + 0

= 2x

Thus, the derivative of x2 is 2x. But it may be difficult to use this limit definition to find the derivatives of complex
functions. Thus, there are some derivative formulas (of course, which are derived from the above limit definition)
that we can use readily in the process of differentiation.

Derivative Formulas in Calculus

The three basic derivatives of the algebraic, logarithmic / exponential and trigonometric functions are derived
from the first principle of differentiation and are used as standard derivative formulas. They are as follows.

Power Rule of Derivatives

By using the above example, the derivative of x2 is 2x. Similarly, we can prove that the derivative of x3 is 3x2, the
derivative of x4 is 4x3, and so on. Power rule generalizes this and it is stated as d/dx (xn) = n xn - 1.

Derivatives of Log/Exponential Functions

The derivative of ln x is, d/dx (ln x) = 1/x

The derivative of log x is, d/dx (loga x) = 1/(x ln a)

The derivative of e^x is, d/dx (ex) = ex

The derivative of a^x is, d/dx (ax) = ax ln a

Derivatives of Trigonometric Functions

Here are the derivatives of trigonometric functions.

If y = cos x, y' = -sin x

If y = tan x, y' = sec2 x

If y = cot x, y' = -cosec2 x

If y = sec x, y' = sec x tan x

If y = cosec x, y' = -cosec x cot x

Derivatives of Inverse Trigonometric Functions

The following are the fundamental rules of derivatives. Let us discuss them in detail.

Power Rule: By this rule, if y = xn , then dy/dx = n x n-1 . Example: d/dx (x5) = 5x4.

Sum/Difference Rule: The derivative process can be distributed over addition/subtraction. i.e., dy/dx [u ± v]=
du/dx ± dv/dx.

Product Rule: The product rule of derivatives states that if a function is a product of two functions, then its
derivative is the derivative of the second function multiplied by the first function added to the derivative of the
first function multiplied by the second function. dy/dx [u × v] = u · dv/dx + v · du/dx. If y = x5 ex , we have y' = x5 .
ex + ex . 5x4 = ex (x5 + 5x4)

Quotient Rule: The quotient rule of derivatives states that d/dx (u/v) = (v · du/dx - u · dv/dx)/ v2

Constant multiple Rule: The constant multiple rule of derivatives states that d/dx [c(f(x)] = c · d/dx f(x). i.e., the
constant which when multiplied by a function, comes out of the differentiation process. For example, d/dx (5x2) =
5 d/dx (x2) = 5(2x) = 10 x.

Constant Rule: The constant rule of derivatives states that the derivative of any constant is 0. If y = k, where k is a
constant, then dy/dx = 0. Suppose y = 4, y' = 0. This rule directly follows from the power rule.

Derivatives of Composite Functions (Chain Rule)

If f and g are differentiable functions in their domain, then f(g(x)) is also differentiable. This is known as the chain
rule of differentiation used for composite functions. (fog)'(x) = f'[(g(x)] g'(x). This also can be written as "if y = f(u)
and u = g(x) , then dy/dx = dy/du · du/dx.

For example, consider y = tan2x. This is a composite function. We can write this function as y = u2, where u = tan x.
Then

dy/du = 2u

du/dx = d/dx (tan x) = sec2x

By the chain rule,

dy/dx = dy/du · du/dx

= 2 tan x sec 2 x

Derivatives of Implicit Functions

In equations where y as a function of x cannot be explicitly defined by the variables x and y, we use implicit
differentiation. If f(x, y) = 0, then differentiate on both sides with respect to x and group the terms containing
dy/dx at one side, and then solve for dy/dx.

For example, 2x + y = 12

d/dx(2x + y) = d/dx(0)

2 + dy/dx = 0

dy/dx = -2

Parametric Derivatives

In a function, we may have the dependent variables x and y which are dependent on the third independent
variable. If x = f(t) and y = g(t), then derivative is calculated as dy/dx = f'(x)/g'(x). Suppose, if x = 4 + t2 and y = 4t2 -
5t4 , then we find dy/dx as follows.

dx/ dt = 2t and dy/dt = 8t -20t3

dy/dx = (dy/dt)/(dx/dt)

dy/dx = (8t -20t3 )/2t

= 2t (4 - 10t2 ) / 2t

dy/dx = 4 - 10t2

Higher-order Derivatives

We can find the successive derivatives of a function and obtain the higher-order derivatives. If y is a function, then
its first derivative is dy/dx. The second derivative is d/dx (dy/dx) which also can be written as d2y/dx2. The third
derivative is d/dx (d2y/dx2) and is denoted by d3y/dx3 and so on.
Alternatively, the first, second, and third derivatives of f(x) can be written as f'(x), f''(x), and f'''(x). For higher order
derivatives, we write the number in brackets as the exponent. Suppose y = 4x3, we get the successive derivatives
as follows. y' = 12x2 , y'' = 24 x and y''' = 24, y(4)= 0.

Partial Derivatives

If u = f(x,y) we can find the partial derivative of with respect to y by keeping x as the constant or we can find the
partial derivative with respect to x by keeping y as the constant. Suppose f(x, y) = x3 y2 , the partial derivatives of
the function are:

∂f/∂x(x3 y2) = 3x2y and

∂f/∂y(x3 y2) = x3 2y

Further, we can find the second-order partial derivatives also like ∂2f/∂x2, ∂2f/∂y2, ∂2f/∂x ∂y, and ∂2f/∂y ∂x.

Finding Derivative Using Logarithmic Differentiation

Sometimes, the functions are too complex to find the derivatives (or) one function might be raised to another
function like y = f(x)g(x). In such cases, we can take log (or) ln on both sides, apply log rules, and then differentiate
on both sides to get dy/dx. This process is known as logarithmic differentiation in calculus.