Question

Use the Chain Rule to find the derivative of the following functions. $$y=(1+2 \tan x)^{15}$$

Short answer

Answer: The derivative of the given function with respect to x is $$\frac{dy}{dx} = 30(1+2 \tan x)^{14}\sec^2 x$$.

Step-by-step solution

Identify the inner and outer functions

First, let's identify the inner and outer functions. In our case, the inner function is $$1+2 \tan x$$, and the outer function is $$u^{15}$$. Let's denote the inner function as $$u$$, so we have: $$u(x)=1+2 \tan x$$ $$y=u^{15}$$ We will use the Chain Rule to find the derivative of y with respect to x.

Apply the Chain Rule

The Chain Rule states that if we have a function $$y$$ which is a function of $$g(x)$$, then the derivative of y is given by: $$\frac{dy}{dx} = \frac{dy}{du} × \frac{du}{dx}$$ where $$y = f(g(x)) = f(u)$$ Here, we need to find two derivatives: $$\frac{dy}{du}$$ (the derivative of $$y$$ with respect to $$u$$, which is the outer function) and $$\frac{du}{dx}$$ (the derivative of $$u$$ with respect to $$x$$, which is the inner function).

Find the derivative of the outer function with respect to u

Let's find $$\frac{dy}{du}$$. Recall that $$y=u^{15}$$. Using the power rule for derivatives, we get: $$\frac{dy}{du} = 15u^{14}$$

Find the derivative of the inner function with respect to x

Now, let's find $$\frac{du}{dx}$$. Recall that $$u(x) = 1+2 \tan x$$, so we need to find the derivative of $$1+2 \tan x$$ with respect to x: $$\frac{du}{dx} = 0 + 2\left(\frac{d}{dx}\tan x\right) = 2\sec^2 x$$

Multiply the two derivatives to find the final derivative

Finally, to find the derivative of the original function, we multiply our calculated derivatives. That is, we compute $$\frac{dy}{dx} = \frac{dy}{du} × \frac{du}{dx}$$: $$\frac{dy}{dx} = (15u^{14})(2\sec^2 x) = 30u^{14}\sec^2 x$$

Substitute the inner function back in the final derivative

Now, we need to replace $$u$$ in the final derivative with the original inner function $$1+2 \tan x$$: $$\frac{dy}{dx} = 30(1+2 \tan x)^{14}\sec^2 x$$ So, the derivative of the given function is: $$\frac{dy}{dx} = 30(1+2 \tan x)^{14}\sec^2 x$$

Key concepts

Derivative

In calculus, a derivative represents the rate at which a function is changing at any given point. Think of it as the function's "speed." When we say we want to "differentiate," we're looking to find the derivative of a given function. This becomes especially important when dealing with functions that are nested inside each other. Using the derivative, we can better understand the behavior of complex functions as they change.
In our exercise, we're asked to find the derivative of a function that involves the chain rule. The chain rule is a formula that helps us take the derivative of composite functions, which are functions within functions. It essentially allows us to "unwrap" these layers one at a time to get the overall rate of change.

Outer Function

When applying the chain rule, identifying the outer function is crucial. The outer function is the larger, overarching function that contains another function inside it. In our example, the outer function is represented by \(u^{15}\). Essentially, you can think of the outer function as the one that would affect the entire expression if you were to change it.

• The outer function affects how the whole chain behaves.
• It's the last part of the function we differentiate directly with respect to the inner part, which is called "u" or any other placeholder variable.

To differentiate this outer function with respect to the inner variable \(u\), we apply the power rule, resulting in the derivative \(\frac{dy}{du} = 15u^{14}\). It's important to note how we "bring down" the exponent in front of the expression to obtain this derivative.

Inner Function

The inner function forms the core part of the composite function. It is the nested function that complicates our differentiation process slightly. In our example, the inner function is \(1 + 2\tan x\). This inner function must be differentiated in addition to the outer function, following the rules of the chain rule.

• To find the derivative of the inner function, we differentiate \(u(x) = 1 + 2\tan x\) with respect to \(x\).
• This yields \(\frac{du}{dx} = 2\sec^2 x\).

One way to visualize this step is as peeling off a layer or "unwrapping" the inner portion of the mathematical expression. Essentially, we're figuring out how the inner part changes with respect to \(x\), which is necessary for using the chain rule correctly.

Power Rule

The power rule is a straightforward but important tool in calculus used for differentiation. It tells us how to take the derivative of a function in the form of \(x^n\). Specifically, if you have a function \(f(x) = x^n\), the derivative \(f'(x)\) is \(nx^{n-1}\).

• In the context of our exercise, the power rule is applied to the outer function \(u^{15}\), giving us \(15u^{14}\).
• Note how in each case, it's about moving the exponent down to become a multiplier before reducing it by one.

Using the power rule simplifies the process of finding the derivative when dealing with polynomial terms. It is one of the most essential rules for differentiating functions, and is especially handy when used in conjunction with the chain rule for nested functions. Together, they provide a systematic approach for untangling complex layers of functions.