# 18 Lecture

## The chain Rule

The chain rule is a fundamental rule of calculus that is used to find the derivative of composite functions. A composite function is a function that is made up of two or more functions.

## Important Mcq's Midterm & Finalterm Prepration Past papers included

What is the chain rule used for in calculus?

A) Integration

B) Derivatives

C) Limits

D) Sequences

Solution: B

Which of the following functions cannot be differentiated using the chain rule?

A) f(x) = sin(x^2)

B) f(x) = e^x + ln(x)

C) f(x) = cos(3x)

D) f(x) = x^2 + x + 1

Solution: D

What is the derivative of f(x) = sin(2x) using the chain rule?

A) 2cos(2x)

B) 2sin(2x)

C) 4cos(2x)

D) 4sin(2x)

Solution: B

What is the derivative of f(x) = e^(3x+2) using the chain rule?

A) 3e^(3x+2)

B) e^(3x+2)

C) 3e^(3x)

D) 2e^(3x+2)

Solution: A

What is the chain rule formula?

A) f'(x) = lim (h->0) (f(x+h)-f(x))/h

B) f(x) = ? g'(x)dx

C) (f(g(x)))' = f'(x)g'(x)

D) (f(g(x)))' = f'(g(x))g'(x)

Solution: D

What is an example of a composite function?

A) f(x) = x^2

B) f(x) = 3x + 4

C) f(x) = sin(x)

D) f(x) = sin(x^2)

Solution: D

Which of the following is the correct order for applying the chain rule?

A) Differentiate the inner function, then the outer function

B) Differentiate the outer function, then the inner function

C) Multiply the inner and outer functions, then differentiate

D) There is no specific order

Solution: B

What is the derivative of f(x) = ln(cos(x)) using the chain rule?

A) -tan(x)

B) -cot(x)

C) -sec(x)

D) -csc(x)

Solution: -tan(x)

Can the chain rule be applied to a function composed of more than two functions?

A) Yes

B) No

Solution: A

Which of the following is a way to remember the chain rule?

A) Outside inside

B) Inside outside

C) Middle first

D) There is no way to remember it

Solution: A

## Subjective Short Notes Midterm & Finalterm Prepration Past papers included

What is the chain rule in calculus?

The chain rule is a rule in calculus that enables us to differentiate composite functions by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

Why do we need the chain rule?

We need the chain rule to differentiate complex functions that are composed of multiple functions. Without the chain rule, it would be challenging to find the derivative of such functions.

What is an example of a composite function?

An example of a composite function is f(g(x)), where f and g are functions of x.

How do we apply the chain rule?

To apply the chain rule, we differentiate the outer function with respect to its variable and multiply it by the derivative of the inner function with respect to its variable.

Can we apply the chain rule to any function?

No, we cannot apply the chain rule to all functions. It only applies to composite functions where one function is nested inside another function.

What is the derivative of sin(x^2)?

The derivative of sin(x^2) is cos(x^2) * 2x.

What is the derivative of e^(3x+2)?

The derivative of e^(3x+2) is 3e^(3x+2).

What is the chain rule formula?

The chain rule formula is (f(g(x)))' = f'(g(x)) * g'(x).

What is the chain rule used for in real-life applications?

The chain rule is used in physics, engineering, and other fields where complex functions are encountered. It is essential in calculating rates of change and gradients of complex systems.

How can one remember the chain rule?

One way to remember the chain rule is to think of it as "outside inside," meaning that we differentiate the outer function first and then the inner function. Another way is to use the mnemonic device "DIDLO," which stands for differentiate the outer function, differentiate the inner function, and multiply.

### The chain Rule

The chain rule is a fundamental rule of calculus that is used to find the derivative of composite functions. A composite function is a function that is made up of two or more functions. The chain rule enables us to find the derivative of such functions. The chain rule states that if y is a function of u, and u is a function of x, then the derivative of y with respect to x is given by the product of the derivative of y with respect to u and the derivative of u with respect to x. This can be written mathematically as: dy/dx = dy/du * du/dx To understand the chain rule better, let us take an example. Suppose we have the composite function f(x) = sin(x^2). To find the derivative of this function, we need to apply the chain rule. Let u = x^2 and y = sin(u). Therefore, the derivative of y with respect to u is cos(u), and the derivative of u with respect to x is 2x. Applying the chain rule, we get: df/dx = dy/du * du/dx = cos(u) * 2x = cos(x^2) * 2x Therefore, the derivative of the composite function f(x) = sin(x^2) is cos(x^2) * 2x. The chain rule can also be applied to more complicated functions. For example, consider the function g(x) = (1 + x^2)^5. To find the derivative of this function, we need to use the chain rule twice. Let u = 1 + x^2 and y = u^5. Therefore, the derivative of y with respect to u is 5u^4, and the derivative of u with respect to x is 2x. Applying the chain rule, we get: dg/dx = dy/du * du/dx = 5u^4 * 2x = 10x(1 + x^2)^4 Therefore, the derivative of the composite function g(x) = (1 + x^2)^5 is 10x(1 + x^2)^4. The chain rule is an essential tool in calculus as it allows us to find the derivative of composite functions. Without the chain rule, we would have to resort to using more complicated methods, such as implicit differentiation, to find the derivative of composite functions. The chain rule also enables us to find the derivative of more complicated functions by breaking them down into simpler functions. In summary, the chain rule is a powerful rule in calculus that is used to find the derivative of composite functions. By breaking down more complicated functions into simpler functions, we can use the chain rule to find the derivative of the composite function. The chain rule is an essential tool in calculus and is used extensively in the study of derivatives and integrals.