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Derivatives of Composite Functions
Derivatives of composite functions can be calculated using the chain rule of differentiation. Let us first recall the meaning of composite functions. Composite functions are functions when a function is written in terms of another function. This implies in a composite function, a function can be substituted into another function and is generally written as (f o g)(x) = f(g(x)). Now, to determine the derivatives of the composite functions, we differentiate the first function with respect to the second function and then differentiate the second function with respect to the variable, i.e., (f o g)'(x) = f'(g(x)). g'(x).
Let us learn how to determine the derivatives of composite functions, the formula to find them, and the concept of partial derivatives of composite functions in two variables with the help of solved examples for a better understanding of the concept.
What are Derivatives of Composite Functions?
Derivatives of composite functions are evaluated using the chain rule method (also known as the composite function rule). The chain rule states that 'Let h be a realvalued function that is a composite of two functions f and g. i.e, h = f o g. Suppose u = g(x), where du/dx and df/du exist, then this could be expressed as:
Derivative of h(x) w.r.t. x = Derivative of f(x) w.r.t. u × Derivative of u w.r.t. x ⇒ d(h(x))/dx = df/du × du/dx
Another way to write the derivatives of composite functions using the chain rule formula is: Derivative of h(x). w.r.t. x = Derivative of f(x) w.r.t. g(x) × Derivative of g(x) w.r.t. x ⇒ d( f(g(x) )/dx = f' (g(x)) · g' (x). In simple words, we say that the derivative of a composite function is the product of the derivative of the outside function with respect to the inside function and the derivative of the inside function with respect to the variable.
Derivatives of Composite Functions Formula
The derivative of a composite function h(x) = f(g(x)) can be determined by taking the product of the derivative of f(x) with respect to g(x) and the derivative of g(x) with respect to the variable x. Mathematically, the formula for the derivatives of composite functions is given as:
Derivatives of Composite Functions In One Variable
Derivatives of composite functions in one variable are determined using the simple chain rule formula. Let us solve a few examples to understand the calculation of the derivatives:
Example 1: Determine the derivative of the composite function h(x) = (x^{3} + 7)^{10}
Solution: Now, let u = x^{3} + 7 = g(x), here h(x) can be written as h(x) = f(g(x)) = u^{10}. So the derivative of h(x) is given by:
d(h(x))/dx = df/du × du/dx
⇒ h'(x) = 10u^{9} × 3x^{2}
= 10(x^{3} + 7)^{9} × 3x^{2}
= 30 x^{2} (x^{3} + 7)^{9}
Example 2: Derivative of composite function y = sin (cos (x^{2}))
Solution: y' = cos(cos (x^{2})). sin (x^{2})). 2x
= 2x sin (x^{2}) cos (cos x^{2})
Partial Derivatives of Composite Functions in Two Variables
Derivative of a function in many variables is calculated with respect to one of the variables at a time. Such derivatives are called partial derivatives. We can calculate the partial derivatives of composite functions z = h(x, y) using the chain rule method of differentiation for one variable. While determining the partial derivative of a function with respect to one variable, we consider all remaining variables as constants. Let us go through an example illustrated below:
Example: Find the x and y derivatives of the composite function f(x, y) = (x^{2}y^{2} + ln x)^{3}
Solution: First, we will differentiate the composite function f(x, y) = (x^{2}y^{2} + ln x)^{3} with respect to x and consider y as a constant.
∂[(x^{2}y^{2} + ln x)^{3}]/∂x = 3 (x^{2}y^{2} + ln x)^{2} × ∂(x^{2}y^{2} + ln x)/∂x
= 3 (x^{2}y^{2} + ln x)^{2} × (2xy^{2} + 1/x)
= 3(2xy^{2} + 1/x)(x^{2}y^{2} + ln x)^{2}
Similarly, we will determine the yderivative considering x as a constant using the chain rule formula.
∂[(x^{2}y^{2} + ln x)^{3}]/∂y = 3 (x^{2}y^{2} + ln x)^{2} × ∂(x^{2}y^{2} + ln x)/∂y
= 3 (x^{2}y^{2} + ln x)^{2} × (2x^{2}y)
= 6x^{2}y (x^{2}y^{2} + ln x)^{2}
Important Notes on Derivatives of Composite Functions
 The tderivative of a composite function z = h(x(t), y(t)) can be calculated using the formula dh/dt = (∂f/∂x) . (dx/dt) + (∂f/∂y) . (dy/dt)
 Derivative of h(x) w.r.t. x = Derivative of f(x) w.r.t. u × Derivative of u w.r.t. x ⇒ d(h(x))/dx = df/du × du/dx, where h(x) = (f o g)(x) and g(x) = u
Related Topics
Derivatives of Composite Functions Examples

Example 1: Calculate the derivative of the composite function f(x) = (1 + sin √x)^{3}
Solution: To find the derivative of f(x), we will use the chain rule method.
f'(x) = 3 (1 + sin √x)^{2} × d(1 + sin √x)/dx [Using dx^{n}/dx = nx^{n1}]
= 3 (1 + sin √x)^{2} × (cos √x) × d(√x)/dx [Using d(sin x)/dx = cos x]
= 3 (1 + sin √x)^{2} × (cos √x) × (1/2√x)
= (3/2) (1/√x) (cos √x) (1 + sin √x)^{2}
Answer: Derivative of f(x) = (1 + sin √x)^{3} is (3/2) (1/√x) (cos √x) (1 + sin √x)^{2}

Example 2: Evaluate the xderivative of h(x, y) = (x^{3}y^{2} + sec xy)^{4} using the partial derivatives of composite functions formula.
Solution: Here, we will determine the partial derivative of h(x, y) w.r.t. x treating y as a constant.
∂h/∂x = 4(x^{3}y^{2} + sec xy)^{3} × ∂(x^{3}y^{2} + sec xy)/∂x
= 4(x^{3}y^{2} + sec xy)^{3} × (3x^{2}y^{2} + sec xy tan xy) × ∂(xy)/∂x
= 4(x^{3}y^{2} + sec xy)^{3} × (3x^{2}y^{2} + sec xy tan xy) × y
= 4y (x^{3}y^{2} + sec xy)^{3} (3x^{2}y^{2} + sec xy tan xy)
Answer: The xderivative of h(x, y) = (x^{3}y^{2} + sec xy)^{4} is 4y (x^{3}y^{2} + sec xy)^{3} (3x^{2}y^{2} + sec xy tan xy)
FAQs on Derivatives of Composite Functions
What are Derivatives of Composite Functions in Calculus?
The derivatives of composite functions are calculated using the chain rule. It is the product of the derivative of the outside function with respect to the inside function and the derivative of the inside function with respect to the variable.
How Do You Use the Chain Rule to Find the Derivative of a Composite Function?
Derivatives of composite functions using the chain rule formula is evaluated as: Derivative of h(x). w.r.t. x = Derivative of f(x) w.r.t. g(x) × Derivative of g(x) w.r.t. x ⇒ d( f(g(x) )/dx = f' (g(x)) · g' (x), where h(x) = (f o g)(x).
How to Find Derivatives of Composite Functions?
The derivatives of composite functions can be determined using the composite function rule (also known as the chain rule method of differentiation).
What are Partial Derivatives of Composite Functions?
We can calculate the partial derivatives of composite functions z = h(x, y) using the chain rule method of differentiation for one variable. While determining the partial derivative of a function with respect to one variable, we consider all remaining variables as constants.
What is the Formula for Derivatives of Composite Functions?
The formula for the differentiation of composite function h(x) = (f o g)(x) is: Derivative of h(x). w.r.t. x = Derivative of f(x) w.r.t. g(x) × Derivative of g(x) w.r.t. x ⇒ d( f(g(x) )/dx = f' (g(x)) · g' (x).
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