Chain Rule Differentiation

Front

What is the Chain Rule in differentiation?

Back

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x).

Front

When should you use the Chain Rule?

Back

You should use the Chain Rule when differentiating a function that is composed of another function, i.e., when there is a function within a function.

Front

Differentiate f(x) = (x^3 - 2x)^2 using the Chain Rule.

Back

f'(x) = 6x^5 - 16x^3 + 8x.

Front

What is the derivative of f(x) = x^7(5 + 8x)^3?

Back

f'(x) = x^6(5 + 8x)^2(35 + 80x).

Front

How do you differentiate a product of functions?

Back

Use the Product Rule: If u(x) and v(x) are functions, then (uv)' = u'v + uv'.

Front

What is the Product Rule in differentiation?

Back

The Product Rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Front

Differentiate f(x) = (9 - 4x^2)^-1 using the Chain Rule.

Back

f'(x) = 8x/(9 - 4x^2)^2.

Front

What is the Quotient Rule in differentiation?

Back

The Quotient Rule states that if you have a function that is the quotient of two functions, u(x) and v(x), then the derivative is (u/v)' = (u'v - uv')/v^2.

Front

What is the significance of the derivative in calculus?

Back

The derivative represents the rate of change of a function with respect to its variable, indicating how the function's output changes as the input changes.

Front

How do you apply the Chain Rule to find the derivative of composite functions?

Back

Identify the outer function and the inner function, differentiate the outer function while keeping the inner function unchanged, then multiply by the derivative of the inner function.