Chain Rule is a way to find the derivative of composite functions. It is one of the basic rules used in mathematics for solving differential problems. It helps us to find the derivative of composite functions such as (3x2 + 1)4, (sin 4x), e3x, (ln x)2, and others. Only the derivatives of composite functions are found using the chain rule. The famous German scientist, Gottfried Leibniz gave the chain rule in the early 17th century....

Derivative of sin inverse x is 1/√(1-x2). The derivative of any function gives the rate of change of the functional value with respect to the input variable. Sin inverse x is one of the inverse trigonometric functions. It is also represented as sin-1x. There are inverse trigonometric functions corresponding to each trigonometric function. The derivative of a function also helps in finding the slope of the tangent to the curve represented by the function at any point....

Derivative of Tan x is sec2x. Derivative of Tan x refers to the process of finding the change in the tangent function with respect to the independent variable. Derivative of tan x is also known as differentiation of tan x....

Derivative of log x is 1/x. Log x Derivative refers to the process of finding change in log x function to the independent variable. The specific process of finding the derivative for log x functions is referred to as logarithmic differentiation. The function log x typically refers to the natural logarithm of x, which is the logarithm to the base e, where e is Euler’s number, approximately equal to 2.71828....

Integral of tan x is ln |sec x| + C. Integral of tan x refers to finding the integration of the trigonometric function tan x with respect to x which can be mathematically formulated as ∫tan x dx. The tangent function, tan x, is an integrable trigonometric function that is defined as the ratio of the sine and cosine functions....

Integration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below....

Implicit Differentiation is a useful tool in the arsenal of tools to tackle problems in calculus and beyond which helps us differentiate the function without converting it into the explicit function of the independent variable. Suppose we don’t know the method of implicit differentiation. In that case, we have to convert each implicit function into an explicit function, which is sometimes very hard and sometimes it is not even possible....

A first-order differential equation is a type of differential equation that involves derivatives of the first degree (first derivatives) of a function. It does not involve higher derivatives. It can generally be expressed in the form: dy/dx = f(x, y). Here, y is a function of x, and f(x, y) is a function that involves x and y....

Derivative of Inverse Trig Function refers to the rate of change in Inverse Trigonometric Functions. We know that the derivative of a function is the rate of change in a function with respect to the independent variable. Before learning this, one should know the formulas of differentiation of Trigonometric Functions. To find the derivative of the Inverse Trigonometric Function, we will first equate the trigonometric function with another variable to find its inverse and then differentiate it using the implicit differentiation formula....

Partial derivatives play a vital role in the area of machine learning, notably in optimization methods like gradient descent. These derivatives help us grasp how a function changes considering its input variables. In machine learning, where we commonly deal with complicated models and high-dimensional data, knowing partial derivatives becomes vital for improving model parameters effectively....

Concavity and points of inflection are the key concepts and basic fundamentals of calculus and mathematical analysis. It provides an insight into how curves behave and the shape of the functions. Where concavity helps us to understand the curving of a function, determining whether it is concave upward or downward, the point of inflection determines the point where the concavity changes, i.e., where either curve transforms from concave upward to concave downward or concave to convex, and vice versa. These concepts are essential in various mathematical applications, including curve sketching, optimization problems , and the study of differential equations....

A tangent plane is a flat surface that touches a curve or surface at a single point, sharing the same slope or direction at that point, facilitating local approximation in calculus. This article discusses tangent planes, which are flat surfaces that touch curves or surfaces at specific points. It explains their definition, how to calculate them, and their geometric interpretation. It also explores their applications in various fields like engineering, physics, and computer graphics....