Find dy/dx y=sin(6x)

Solution:

Step 1:

Apply the derivative operator to both sides of the equation: $\frac{d}{dx}(y)=\frac{d}{dx}(\sin (6x))$

Step 2:

The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.

Step 3:

Take the derivative of the right-hand side.

Step 3.1:

Use the chain rule for differentiation, which is expressed as $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$, where $f(x)=\sin (x)$ and $g(x)=6x$.

Step 3.1.1:

Identify $u=6x$ and then differentiate $\sin (u)$ with respect to $u$ and $6x$ with respect to $x$: $\frac{d}{du}[\sin (u)]\cdot \frac{d}{dx}[6x]$

Step 3.1.2:

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$: $\cos (u)\cdot \frac{d}{dx}[6x]$

Step 3.1.3:

Substitute $6x$ back in place of $u$: $\cos (6x)\cdot \frac{d}{dx}[6x]$

Step 3.2:

Perform the differentiation.

Step 3.2.1:

Recognize that $6$ is a constant and differentiate $6x$ with respect to $x$: $\cos (6x)\cdot (6\cdot \frac{d}{dx}[x])$

Step 3.2.2:

Apply the power rule, which states that the derivative of $x^n$ is $n{x}^{n-1}$, where $n=1$: $\cos (6x)\cdot (6\cdot 1)$

Step 3.2.3:

Simplify the derivative.

Step 3.2.3.1:

Multiply $6$ by $1$: $\cos (6x)\cdot 6$

Step 3.2.3.2:

Rearrange the terms to place the constant before the trigonometric function: $6\cdot \cos (6x)$

Step 4:

Combine the results to form the derivative equation: $\frac{dy}{dx}=6\cdot \cos (6x)$

Step 5:

Express the final derivative: $\frac{dy}{dx}=6\cos (6x)$

Knowledge Notes:

The problem involves finding the derivative of a trigonometric function composed with a linear function, which is a common calculus task. The key knowledge points involved in solving this problem include:

1. Derivative of a Function: The derivative represents the rate at which a function is changing at any given point. For a function $y=f(x)$, the derivative with respect to $x$ is denoted as $\frac{dy}{dx}$ or $f^{\prime }(x)$.

2. Chain Rule: A rule for computing the derivative of the composition of two or more functions. If $y=f(g(x))$, then the derivative of $y$ with respect to $x$ is $\frac{dy}{dx}=f^{\prime }(g(x))\cdot g^{\prime }(x)$.

3. Derivative of Sine Function: The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$.

4. Derivative of a Constant Multiplied by a Function: If $y=c\cdot f(x)$ where $c$ is a constant, then the derivative is $\frac{dy}{dx}=c\cdot f^{\prime }(x)$.

5. Power Rule: A basic rule of differentiation. If $y=x^n$, then the derivative of $y$ with respect to $x$ is $\frac{dy}{dx}=n\cdot x^{n-1}$.

6. Simplification: After applying the chain rule and the power rule, it's important to simplify the expression to its most basic form, which often involves multiplying constants or rearranging terms for clarity.

These concepts are foundational in calculus and are frequently used in various applications, including physics, engineering, and economics.