Derivative of Composite Function
Composite Function is defined as the function of a function. Let’s say we have function f which is a function of another function g(x) then the composite function is written as f(g(x)) or fog(x). Let’s say we have a function y = sin2x then we can find the derivative in the following manner:
Step 1: First assume one function equal to some other variable i.e. u = sin x (let). There for original function becomes y = u2 where u = sin x
Step 2: Now derivative of u = sin x with respect to x and y = u2 with respect to u. Therefore we have du/dx = cos x and dy/du = 2u
Step 3: Now multiply the two derivatives i.e. (du/dx)(dy/du) = 2u cos x.
Step 4: Now replace the assumed value u = sin x from step 1.
Hence, the derivative of sin2x is 2sin x cos x.
Chain Rule of Derivatives
Chain Rule allows us to differentiate Composite Functions in a single line. In the chain rule, we differentiate functions and write them in product format. For Example if f(x) = sin2x then f'(x) = 2sin x cos x. In this example we first differentiated sin2x to 2sin x using the Power Rule of Derivative and then differentiated sin x to cos x and wrote both derivatives in product format.
Derivatives | First and Second Order Derivatives, Formulas and Examples
Derivatives: In mathematics, a Derivative represents the rate at which a function changes as its input changes. It measures how a function’s output value moves as its input value nudges a little bit. This concept is a fundamental piece of calculus. It is used extensively across science, engineering, economics, and more to analyze changes.
Table of Content
- What are Derivatives?
- Derivatives Meaning
- Derivative by First Principle
- Types of Derivatives
- First Order Derivative
- Second Order Derivative
- nth Order Derivative
- Derivatives Formula
- Rules of Derivatives
- Derivative of Composite Function
- Chain Rule of Derivatives
- Derivative of Implicit Function
- Parametric Derivatives
- Higher Order Derivatives
- Partial Derivative
- Logarithmic Differentiation
- Applications of Derivatives
- Derivatives Examples
- Sample Problems on Derivatives
- Practice Problems on Derivatives
A derivative is a calculus tool that measures the sensitivity of a function’s output to its input. It is also known as the instantaneous rate of change of a function at a given position.
The derivative of a function with just one variable is the slope of the line that is tangent to the function’s graph for a given input value. In terms of geometry, the derivative of a function can be defined as the slope of its graph.
In this article we have covered Meaning of Derivatives along with types of derivatives, examples, formulas, applications and many more.
Contact Us