Chain Rule & 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

What is implicit differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function that is not explicitly solved for one variable in terms of another. It involves differentiating both sides of an equation with respect to a variable and then solving for the derivative.

Front

Find the derivative of y = 5x^2 e^(3x).

Back

y' = 5xe^(3x)(3x + 2).

Front

What does the derivative of a function represent?

Back

The derivative of a function represents the instantaneous rate of change of the function with respect to its variable, or the slope of the tangent line to the graph of the function at a given point.

Front

Find the equation of the tangent line to the graph of f(x) at the point (1, 100) for f(x) = (5x^5 + 5)^2.

Back

y = 500x - 400.

Front

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

Back

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

Front

What is the relationship between the derivative and the slope of a tangent line?

Back

The derivative at a point gives the slope of the tangent line to the graph of the function at that point.

Front

What is the formula for the derivative of a product of two functions?

Back

If u(x) and v(x) are functions, then the derivative of their product is given by (uv)' = u'v + uv'.

Front

What is the formula for the derivative of a quotient of two functions?

Back

If u(x) and v(x) are functions, then the derivative of their quotient is given by (u/v)' = (u'v - uv')/v^2.

Front

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

Back

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