Category: Calculus

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 let us find the parametric derivative. dx/ dt…

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 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…

The following are the fundamental rules of derivatives. Let us discuss them in detail. Power Rule: The power rule of derivatives states that if a function is an algebraic expression raised to any power, say n, then the derivative has a power 1 less than the original function.  If y = xn , where n > 0. Then dy/dx…

If f and g are the derivative 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 y = f(u) and t = g(x) , then dy/dx = dy/ dt . dt/dx For example, consider y…

Here are the derivatives of inverse trigonometric functions. If y = sin-1 x, y’ = 1 / √(1−x2) If y = cos-1 x, y’ = −1 / √(1−x2) If y = tan-1 x, y’ = 1/ (1+x2) If y = cot-1 x, y’ =−1/(1+x2) If y = sec-1 x, y’ = 1 / x√(x2−1) If y = cosec-1 x, y’ = −1 / x√(x2−1)

Here are the derivatives of trigonometric functions. If y = sin x, y’ = cos x If y = cos x, y’ = -sin x If y = tan x, y’ = sec2 x If y = sec x, y’ = sec x tan x If y = cosec x, y’ = -cosec x cot x If…

The three basic derivatives of the algebraic, logarithmic/ exponential and trigonometric functions are derived from the fundamental principle of differentiation and are used as standard derivative formulas. They are as follows: If y = lnee x, then dy/dx = 1/x if y = logaa x, then dy/dx = 1/[(log a) x] If y = a x , dy/dx= ax log a

Derivatives are obtained by applying the limits as per the first principle of differentiation that we obtained as the definition of a derivative. Let f(x) = 4×2 + 3 f′(x)=lim δx→0 f(x+δx) − f(x) / δx f ′ ( x ) = lim δ x → 0 f ( x + δ x ) − f ( x…

The derivative is a function that is geometrically defined as the slope of the line tangent to the curve at any point. If f is differentiable and continuous at [a,b], then This change in h is infinitely very small. We denote it by Δx. Then change in the original function f(x) is also small, denoted…

A derivative is the rate of change of a quantity y with respect to another quantity x. A derivative is also termed the differential coefficient of y with respect to x. Differentiation is the process of finding the derivative of a function. If f(x) is a function differentiable in an interval [a,b], at every point of the…