chain rule derivatives practice

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 two functions f(x) and g(x), then the derivative of their composition f(g(x)) is f'(g(x)) * g'(x).

Front

How do you find the derivative of a function raised to a power, such as f(x) = (g(x))^n?

Back

To find the derivative of a function raised to a power, use the chain rule: f'(x) = n * (g(x))^(n-1) * g'(x).

Front

What is the derivative of f(x) = x^n?

Back

The derivative of f(x) = x^n is f'(x) = n * x^(n-1).

Front

How do you differentiate a quotient of two functions, such as y = u/v?

Back

To differentiate a quotient, use the quotient rule: y' = (v * u' - u * v') / v^2.

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 + uv'.

Front

How do you find the slope of the tangent line to a curve at a given point?

Back

To find the slope of the tangent line to a curve at a given point, calculate the derivative of the function at that point.

Front

What is the significance of the derivative in relation to a function's graph?

Back

The derivative represents the rate of change of the function and the slope of the tangent line at any point on the graph.

Front

How do you apply the chain rule to find the derivative of y = (x^2 + 1)^3?

Back

Let u = x^2 + 1, then y = u^3. The derivative is dy/dx = 3u^2 * (2x) = 3(x^2 + 1)^2 * (2x).

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

How do you find the derivative of a rational function, such as y = (x + 2)/(x - 3)?

Back

Use the quotient rule: y' = [(x - 3)(1) - (x + 2)(1)] / (x - 3)^2.