Product/Quotient/Chain Rule Review

Front

What is the Product Rule in differentiation?

Back

The Product Rule states that if you have two functions, f(x) and g(x), then the derivative of their product h(x) = f(x)g(x) is given by h'(x) = f'(x)g(x) + f(x)g'(x).

Front

What is the Quotient Rule in differentiation?

Back

The Quotient Rule states that if you have two functions, f(x) and g(x), then the derivative of their quotient h(x) = f(x)/g(x) is given by h'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2.

Front

What is the Chain Rule in differentiation?

Back

The Chain Rule states that if you have a composite function h(x) = f(g(x)), then the derivative is h'(x) = f'(g(x))g'(x).

Front

Differentiate the function h(x) = f(x)g(x) at x = 1 using the Product Rule.

Back

h'(1) = f'(1)g(1) + f(1)g'(1).

Front

Find the derivative of y = 2x^2(3x - 4) using the Product Rule.

Back

y' = 2x^2(3) + (2)(3x - 4)(2x) = 18x^2 - 16x.

Front

Differentiate the function (2x + 2)(x - 5)^8 using the Product Rule and Chain Rule.

Back

The derivative is 8(2x + 2)(x - 5)^7 + 2(x - 5)^8.

Front

Find the derivative of f(t) = (t^2 + 2t)^5 using the Chain Rule.

Back

f'(t) = 5(t^2 + 2t)^4(2t + 2).

Front

Differentiate f(x) = 5sin(x) + 3x^3cos(x) using the Product Rule.

Back

f'(x) = 5cos(x) - 3x^3sin(x) + 9x^2cos(x).

Front

What is the derivative of a constant function?

Back

The derivative of a constant function is 0.

Front

If f(x) = x^n, what is f'(x)?

Back

f'(x) = nx^{n-1}.