Category: 2. Derivatives

A derivative of a function is the rate of change of one quantity over the other. Derivative of any continuous function that is differentiable at [a,b] is derived using the first principle of differentiation using the limits. If f(x) is given,

If u=f(x,y) we can find the partial derivative of y keeping x as the constant or we can find the partial derivative of x by keeping y as the constant. Suppose f(x,y) = x3 y2 , the partial derivatives of the function are: ?f/dx(x3 y2) = 3x2y and ?f/dy(x3 y2) = x3 2y

We can find the successive derivatives of a function and obtain the higher-order derivatives. If y is a function, then its first derivtive is dy/dx. The second derivative is  d / dx . dy / dx The third derivative is  d / dx . d2y / dx2 and so on. Suppose y = 4×3 , we… Continue reading Higher-order 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 let us find the parametric derivative. dx/ dt… Continue reading Parametric Derivatives

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… Continue reading Derivatives of Implicit Functions

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… Continue reading Fundamental Rules of Derivatives

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… Continue reading Derivatives of Composite Functions

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… Continue reading Derivatives of Trigonometric Functions

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