Tangents Suppose that u=g(x) is differentiable at x=1 and...

Tangents Suppose that $u=g(x)$ is differentiable at $x=1$ and that $y=f(u)$ is differentiable at $u=g(1)$ . If the graph of $y=f(g(x))$ has a horizontal tangent at $x=1,$ can we conclude anything about the tangent to the graph of $g$ at $x=1$ or the tangent to the graph of $f$ at $u=g(1) ?$ Give reasons for your answer.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Chain Rule

The chain rule is a formula used to compute the derivative of a composite function. It states that if a function y = f(u) is composed with u = g(x), then the derivative of y with respect to x is given by f'(g(x)) multiplied by g'(x). This concept is essential in understanding how changes in the inner function affect the outer function.

Composite Function Differentiability

For a composite function to be differentiable, both the inner function and the outer function must be differentiable at the appropriate points. This ensures that the chain rule can be applied, and the derivative of the composite function exists.

Tangent Lines and Their Slopes

The slope of a tangent line to a curve at a given point is defined by the derivative of the function at that point. When the tangent line is horizontal, its slope is zero, indicating that the derivative of the function at that point is zero.

Horizontal Tangents

A horizontal tangent implies that the derivative of the function at that point is zero. In the context of composite functions, if the composite function has a horizontal tangent, the product of the derivatives from the chain rule must equal zero, meaning at least one of the derivatives must vanish.

Please give Ace some feedback