Find dy/dx y=x^(sin(x))

Solution:

Step 1:

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}(x^{\sin(x)})$

Step 2:

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

Step 3:

Proceed to differentiate the right-hand side.

Step 3.1:

Utilize logarithmic properties to facilitate differentiation.

Step 3.1.1:

Express $x^{\sin(x)}$ as $e^{\ln(x^{\sin(x)})}$: $\frac{d}{dx}[e^{\ln(x^{\sin(x)})}]$

Step 3.1.2:

Transform $\ln(x^{\sin(x)})$ by bringing $\sin(x)$ in front of the log: $\frac{d}{dx}[e^{\sin(x)\ln(x)}]$

Step 3.2:

Apply the chain rule, which states that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$, where $f(x) = e^x$ and $g(x) = \sin(x)\ln(x)$.

Step 3.2.1:

Introduce $u = \sin(x)\ln(x)$ to apply the chain rule: $\frac{d}{du}[e^u]\frac{d}{dx}[\sin(x)\ln(x)]$

Step 3.2.2:

Differentiate using the exponential rule, which says the derivative of $a^u$ is $a^u\ln(a)$ where $a=e$: $e^u\frac{d}{dx}[\sin(x)\ln(x)]$

Step 3.2.3:

Substitute $u$ back with $\sin(x)\ln(x)$: $e^{\sin(x)\ln(x)}\frac{d}{dx}[\sin(x)\ln(x)]$

Step 3.3:

Apply the product rule, which says the derivative of $f(x)g(x)$ is $f(x)g'(x) + g(x)f'(x)$, where $f(x) = \sin(x)$ and $g(x) = \ln(x)$: $e^{\sin(x)\ln(x)}(\sin(x)\frac{d}{dx}[\ln(x)] + \ln(x)\frac{d}{dx}[\sin(x)])$

Step 3.4:

The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$: $e^{\sin(x)\ln(x)}(\sin(x)\frac{1}{x} + \ln(x)\frac{d}{dx}[\sin(x)])$

Step 3.5:

Combine $\sin(x)$ and $\frac{1}{x}$: $e^{\sin(x)\ln(x)}\left(\frac{\sin(x)}{x} + \ln(x)\frac{d}{dx}[\sin(x)]\right)$

Step 3.6:

The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$: $e^{\sin(x)\ln(x)}\left(\frac{\sin(x)}{x} + \ln(x)\cos(x)\right)$

Step 3.7:

Simplify the expression.

Step 3.7.1:

Apply the distributive property: $e^{\sin(x)\ln(x)}\frac{\sin(x)}{x} + e^{\sin(x)\ln(x)}(\ln(x)\cos(x))$

Step 3.7.2:

Combine $e^{\sin(x)\ln(x)}$ and $\frac{\sin(x)}{x}$: $\frac{e^{\sin(x)\ln(x)}\sin(x)}{x} + e^{\sin(x)\ln(x)}\ln(x)\cos(x)$

Step 3.7.3:

Reorder the terms for clarity: $e^{\sin(x)\ln(x)}\ln(x)\cos(x) + \frac{e^{\sin(x)\ln(x)}\sin(x)}{x}$

Step 4:

Formulate the equation by equating the left and right sides: $y = e^{\sin(x)\ln(x)}\ln(x)\cos(x) + \frac{e^{\sin(x)\ln(x)}\sin(x)}{x}$

Step 5:

Replace $y$ with $\frac{dy}{dx}$ to get the final derivative: $\frac{dy}{dx} = e^{\sin(x)\ln(x)}\ln(x)\cos(x) + \frac{e^{\sin(x)\ln(x)}\sin(x)}{x}$

Knowledge Notes:

To solve the given problem, we need to apply several calculus rules and properties:

1. Chain Rule: This rule is used for differentiating compositions of functions. It states that $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$.

2. Product Rule: This rule is used when differentiating products of functions. It states that $\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$.

3. Exponential Rule: This rule is used when differentiating exponential functions. It states that $\frac{d}{dx}[a^x] = a^x\ln(a)$, where $a$ is a constant.

4. Logarithmic Properties: These properties help to simplify expressions involving logarithms. For example, $\ln(x^y) = y\ln(x)$.

5. Derivatives of Basic Functions: The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$, and the derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$.

By applying these rules and properties systematically, we can find the derivative of the given function $y = x^{\sin(x)}$.