Derivatives the Chain Rule

Front

What is the Chain Rule in calculus?

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' is given by y' = f'(g(x)) * g'(x).

Front

How do you differentiate a function raised to a power, such as y = (u(x))^n?

Back

To differentiate a function raised to a power, use the power rule in conjunction with the Chain Rule: y' = n * (u(x))^(n-1) * u'(x), where u(x) is a function of x.

Front

What is the derivative of cos(x)?

Back

The derivative of cos(x) is -sin(x).

Front

How do you find the derivative of a composite function?

Back

To find the derivative of a composite function, apply the Chain Rule: differentiate the outer function and multiply it by the derivative of the inner function.

Front

What is the power rule for differentiation?

Back

The power rule states that if f(x) = x^n, then f'(x) = n * x^(n-1).

Front

What is the derivative of sin(x)?

Back

The derivative of sin(x) is cos(x).

Front

How do you differentiate a product of two functions, such as y = u(x)v(x)?

Back

To differentiate a product of two functions, use the product rule: y' = u'v + uv', where u and v are functions of x.

Front

What is the derivative of f(x) = (x^6 + 4)^5?

Back

Using the Chain Rule, the derivative is f'(x) = 30x^5(x^6 + 4)^4.

Front

What is the derivative of f(x) = cos^5(4x^3)?

Back

Using the Chain Rule, the derivative is f'(x) = -60x^2cos^4(4x^3)sin(4x^3).

Front

What is the product rule for differentiation?

Back

The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is (u*v)' = u'v + uv'.