Problem 15
Question
Use implicit differentiation to find \(d y / d x\). $$e^{2 x}=\sin (x+3 y)$$
Step-by-Step Solution
Verified Answer
\( \frac{dy}{dx} = \frac{2e^{2x} - \cos(x + 3y)}{3\cos(x + 3y)} \).
1Step 1: Implicit Differentiation
Starting with the equation \( e^{2x} = \sin(x + 3y) \), differentiate both sides with respect to \( x \). For the left side, use the chain rule: differentiate \( e^{2x} \) to get \( 2e^{2x} \).
2Step 2: Chain Rule Application
Differentiate \( \sin(x + 3y) \) with respect to \( x \). Use the chain rule: the derivative of \( \sin(u) \) is \( \cos(u) \), where \( u = x + 3y \), and then multiply by the derivative of \( u \) with respect to \( x \). This becomes \( \cos(x + 3y) (1 + 3 \frac{dy}{dx}) \) because \( \frac{d}{dx}(3y) = 3 \cdot \frac{dy}{dx} \).
3Step 3: Equate Derivatives
Equate the derivatives from both sides to obtain: \[ 2e^{2x} = \cos(x + 3y) (1 + 3 \frac{dy}{dx}) \].
4Step 4: Solve for \( \frac{dy}{dx} \)
First, distribute \( \cos(x + 3y) \): \[ 2e^{2x} = \cos(x + 3y) + 3\cos(x + 3y)\frac{dy}{dx} \].Isolate terms involving \( \frac{dy}{dx} \):\[ 2e^{2x} - \cos(x + 3y) = 3\cos(x + 3y)\frac{dy}{dx} \].
5Step 5: Isolate \( \frac{dy}{dx} \)
Solve for \( \frac{dy}{dx} \) by dividing both sides by \( 3\cos(x + 3y) \):\[ \frac{dy}{dx} = \frac{2e^{2x} - \cos(x + 3y)}{3\cos(x + 3y)} \].
Key Concepts
Chain RuleDifferentiationTrigonometric Functions
Chain Rule
The Chain Rule is one of the fundamental concepts in calculus. It is essential when dealing with compositions of functions, allowing us to differentiate functions within functions. In our exercise, we encounter it when differentiating the expression \( \sin(x + 3y) \) with respect to \( x \).
Here's how it works: when you differentiate a function such as \( \sin(u) \), where \( u \) is itself a function of \( x \), you apply the chain rule. First, find the derivative of the outer function, \( \sin(u) \), which is \( \cos(u) \).
Next, multiply that by the derivative of the inner function \( u = x + 3y \). Therefore, the derivative of \( u \) with respect to \( x \) involves the terms \( 1 \) for the derivative of \( x \) and \( 3 \frac{dy}{dx} \) for \( 3y \). The chain rule unboxed is:
Here's how it works: when you differentiate a function such as \( \sin(u) \), where \( u \) is itself a function of \( x \), you apply the chain rule. First, find the derivative of the outer function, \( \sin(u) \), which is \( \cos(u) \).
Next, multiply that by the derivative of the inner function \( u = x + 3y \). Therefore, the derivative of \( u \) with respect to \( x \) involves the terms \( 1 \) for the derivative of \( x \) and \( 3 \frac{dy}{dx} \) for \( 3y \). The chain rule unboxed is:
- Differentiate the outer function
- Multiply by the derivative of the inner function
Differentiation
Differentiation is the process of finding the derivative, which measures how a function changes as its input changes. In problems involving implicit differentiation, like ours, both sides of an equation are differentiated with respect to a given variable—usually \( x \).
When differentiating, each side of the equation may involve more than one function of \( x \), including products, quotients or composed functions. This necessitates the application of various rules, such as the product, quotient, and importantly for this exercise, the chain rule. Let's look closely at the left-hand side of our equation \( e^{2x} \).
To differentiate \( e^{2x} \), acknowledge that it involves an exponential function where the exponent itself is a function of \( x \). Apply the chain rule to find its derivative. Differentiate the exponent function \( 2x \) to obtain \( 2 \), and multiply this by the original function \( e^{2x} \), resulting in \( 2e^{2x} \).
When differentiating, each side of the equation may involve more than one function of \( x \), including products, quotients or composed functions. This necessitates the application of various rules, such as the product, quotient, and importantly for this exercise, the chain rule. Let's look closely at the left-hand side of our equation \( e^{2x} \).
To differentiate \( e^{2x} \), acknowledge that it involves an exponential function where the exponent itself is a function of \( x \). Apply the chain rule to find its derivative. Differentiate the exponent function \( 2x \) to obtain \( 2 \), and multiply this by the original function \( e^{2x} \), resulting in \( 2e^{2x} \).
- Start with differentiating the whole function as a whole
- Repeat with each sub-part that is independently a function of \( x \)
Trigonometric Functions
Trigonometric functions are vital in calculus and frequently appear in differentiation problems. In this exercise, we deal with \( \sin(x + 3y) \) which involves implicit differentiation.
The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \). This derivative often becomes more intricate when \( u \) itself is a composed or implicit function of another variable, like \( x \). Here, \( u = x + 3y \) means we must further differentiate \( u \) with respect to \( x \) using the chain rule mentioned earlier.
Remember trigonometric derivatives:
The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \). This derivative often becomes more intricate when \( u \) itself is a composed or implicit function of another variable, like \( x \). Here, \( u = x + 3y \) means we must further differentiate \( u \) with respect to \( x \) using the chain rule mentioned earlier.
Remember trigonometric derivatives:
- \( \frac{d}{dx}(\sin(x)) = \cos(x) \)
- \( \frac{d}{dx}(\cos(x)) = -\sin(x) \)
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