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 $f^{\prime }(g(x))g^{\prime }(x)$, 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 $f(x)g^{\prime }(x)+g(x)f^{\prime }(x)$, 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 $\frac{1}{x}$: $e^{\sin (x)\ln (x)}(\frac{\sin (x)}{x}+\ln (x)\frac{d}{dx}[\sin (x)])$

Step 3.6:

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

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 $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))g^{\prime }(x)$.

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

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 $y=x^{\sin (x)}$.