Problem 12

Question

In Problems 1-18, find $D_{x} y$. $$ y=\sin x \tan x $$

Step-by-Step Solution

Verified

Answer

The derivative is $D_{x} y = \cos x \tan x + \sin x \sec^2 x$.

1Step 1: Understand the Problem

The exercise asks us to differentiate the function $y = \sin x \tan x$ with respect to $x$.

2Step 2: Apply the Product Rule

We have the product of two functions: $u = \sin x$ and $v = \tan x$. According to the product rule, the derivative $D_{x}(uv)$ is $u'v + uv'$.

3Step 3: Differentiate $u = \sin x$

Find the derivative of $u$:$\frac{d}{dx}(\sin x) = \cos x$.

4Step 4: Differentiate $v = \tan x$

Find the derivative of $v$:$\frac{d}{dx}(\tan x) = \sec^2 x$.

5Step 5: Substitute into Product Rule Formula

Substitute $u$, $v$, $u'$, and $v'$ into the product rule formula:$D_{x}(\sin x \tan x) = (\cos x)(\tan x) + (\sin x)(\sec^2 x)$.

6Step 6: Simplify the Expression

Simplify the expression:$D_{x}(\sin x \tan x) = \cos x \tan x + \sin x \sec^2 x$.No further simplification is possible, as this is the expression for the derivative.

Key Concepts

Product RuleTrigonometric FunctionsDerivative of Trigonometric Functions

Product Rule

When dealing with the differentiation of a product of two functions, the product rule is a fundamental tool. It allows us to tackle these problems with ease. The product rule states that if you have a function built by multiplying two functions, say $u(x)$ and $v(x)$, their derivative will be the sum of two products: one where you differentiate $u$ and multiply by $v$, and another where you take $u$ as is and multiply by the derivative of $v$:

$D_x(uv) = u'v + uv'$

In our example of $y = \sin x \tan x$, we treat $\sin x$ as $u(x)$ and $\tan x$ as $v(x)$. Applying the product rule will help us find the derivative efficiently, by breaking down the problem into manageable steps. Whenever you find yourself with a product of functions in a derivative problem, remember: the product rule is your best friend!

Trigonometric Functions

Trigonometric functions are crucial in calculus, especially when dealing with problems involving angles and periodic phenomena. The primary trigonometric functions, sine ($\sin$), cosine ($\cos$), and tangent ($\tan$), are the basis for understanding their behavior in calculus.

$\sin x$ represents the opposite side over the hypotenuse in a right triangle.
$\cos x$ is the adjacent side over the hypotenuse.
$\tan x$ is the ratio of $\sin x$ to $\cos x$ or opposite over adjacent.

These functions, along with their related functions like secant ($\sec x$), are periodic and have properties that make them go around the circle of angles seamlessly. We use these properties as we solve calculus problems: the sine and cosine functions are particularly important as they derive into each other, forming a cycle of derivation.

Derivative of Trigonometric Functions

Understanding the derivatives of trigonometric functions is vital in calculus. Each trigonometric function has a specific derivative:

The derivative of $\sin x$ is $\cos x$.
The derivative of $\cos x$ is $-\sin x$.
The derivative of $\tan x$ is $\sec^2 x$.

These derivatives are crucial each time you need to find how trigonometric functions change in respect to another variable, like $x$ in our problem. For example, in deriving $y = \sin x \tan x$, we use the derivatives of $\sin x$ and $\tan x$ to apply the product rule effectively. Knowing these derivatives by heart will significantly aid in quickly solving any calculus problem involving trigonometric functions.

Problem 11

Problem 12

Other exercises in this chapter

Problem 11

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=x^{2}+2 x $$

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Problem 11

In Problems 1-20, find $D_{x} y$. $$ y=\cos ^{3} x $$

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Problem 12

In Problems 9-16, find $f^{\prime \prime}(2)$. $$ f(u)=\frac{2 u^{2}}{5-u} $$

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Problem 12

Find $D_{x} y$. $$ y=\sinh x \cosh 4 x $$

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