Consider the functions f(u)=u^q and g(x)=x^p / q. Assume...

Consider the functions $f(u)=u^{q}$ and $g(x)=x^{p / q}$. Assume that $g$ is differentiable. (a) Show that $f(g(x))=x^{p}$ (recall the Laws of Exponents). (b) Apply the Chain Rule and the Power Rule for whole-number exponents to show that $f^{\prime}(g(x)) g^{\prime}(x)=p x^{p-1}$. (c) Use the result of (b) to derive the Power Rule for $x^{p / q}$.

Consider the functions $f(u)=u^{q}$ and $g(x)=x^{p / q}$. Assume that $g$ is differentiable.
(a) Show that $f(g(x))=x^{p}$ (recall the Laws of Exponents).
(b) Apply the Chain Rule and the Power Rule for whole-number exponents to show that $f^{\prime}(g(x)) g^{\prime}(x)=p x^{p-1}$.
(c) Use the result of (b) to derive the Power Rule for $x^{p / q}$.

Key Concepts

Power Rule

The Power Rule is a basic rule in differential calculus that provides a straightforward method for finding the derivative of power functions of the form x^n, where n is any real number. According to this rule, the derivative of x^n is given by n times x raised to the power (n-1), making it an essential tool for differentiating polynomial functions and expressions involving exponents.

Differentiation of Composite Functions with Exponents

Differentiating composite functions that contain exponents involves applying both the Chain Rule and the Power Rule. In such cases, the function is first broken down into an outer power function and an inner function, allowing the derivative to be computed as the product of the derivative of the outer function (using the Power Rule) evaluated at the inner function, and the derivative of the inner function. This approach is particularly useful when dealing with functions with rational or fractional exponents.

Laws of Exponents

The Laws of Exponents are a set of rules that describe how to handle operations on expressions with exponents, such as multiplying powers by adding exponents, raising a power to a power by multiplying exponents, and rewriting expressions in simplified form. These laws are essential in simplifying and transforming functions, especially when composing them or comparing their forms.

Chain Rule

The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. It states that if one function is composed with another, the derivative of the composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. This rule is key in differentiating more complex functions that are built from simpler, nested functions.

Transcript

00:01 All right, so we've got a couple of functions, f and g.

00:04 And the first couple problems aren't too bad.

00:06 So we're going to just find the composition.

00:09 So that means we're taking f of g.

00:13 So we're putting in this thing for you.

00:16 So we're going to get x to the p over q to the power of q.

00:20 Because that thing goes in for you.

00:23 And now exponent rule says the exponents multiply, so that's x to the p like they wanted.

00:28 Okay.

00:29 And now, if we look at this thing, that's the expression for the chain rule, right? that's the derivative of f of whatever's inside times the derivative of the inside.

00:37 So the derivative of that must be the derivative of x to the p, and that's an easy power rule.

00:42 That's p times x the p minus one.

00:46 Okay, and now part c is the one that's worded kind of weirdly.

00:49 They want you to find a power rule for x to the p over q using what you got in part b.

00:56 So what does that mean? they want you to find a rule for taking the derivative of x the p over q, which in fact is g of x.

01:05 So really we're looking for this.

01:09 Well, we know what the whole thing is.

01:11 So really, if we could just find this highlighted thing, we'd be good.

01:15 Okay.

01:16 But what is that? well, f of u is that...