FLASHCARD - Chain Rule and Implicit Differentiation

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' = f'(g(x)) * g'(x).

Front

Differentiate y = sin(2x^2).

Back

y' = 4x cos(2x^2).

Front

What is Implicit Differentiation?

Back

Implicit Differentiation is a technique used to differentiate equations that define y implicitly in terms of x, rather than explicitly as y = f(x).

Front

Find dy/dx for the equation 9y + x = 4y.

Back

dy/dx = -1/5.

Front

Differentiate y = e^(5x) + 1.

Back

y' = 5e^(5x).

Front

What is the product rule in differentiation?

Back

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

Front

Differentiate f(x) = x^7 (5 + 8x)^3.

Back

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

Front

What is the Quotient Rule in calculus?

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 given by (u/v)' = (u'v - uv')/v^2.

Front

Differentiate y = sin^3(4x^-6).

Back

y' = -24x^-7 * cos(4x^-6) * 3sin^2(4x^-6).

Front

What is the derivative of e^x?

Back

The derivative of e^x is e^x.