Question

What is the differentiation of ( ln(sin(x)ˣ) )?

a) ( 1/sin(x) )
b) ( cos(x)/sin(x)ˣ )
c) ( xln(sin(x)) )
d) ( cos(x) + xln(sin(x)) )

Answer 1

The differentiation of ln(sin(x)ˣ) can be found using the chain rule. Start by applying the chain rule to find the derivative, and then break down the function into smaller parts. Finally, put all the parts together to find the complete differentiation.

Step-by-step explanation:

The differentiation of ln(sin(x)ˣ) can be found using the chain rule. Let's break it down step by step:

1. Start by applying the chain rule. The derivative of ln(u) is (1/u) * du/dx, where u is the function inside the natural logarithm.
2. In this case, u = sin(x)ˣ. Now we need to find du/dx.
3. Using the property of exponentiation, we can rewrite sin(x)ˣ as e^(xln(sin(x))).
4. Now, we can find du/dx by applying the chain rule again. The derivative of e^(xln(sin(x))) is e^(xln(sin(x))) * d/dx(xln(sin(x))).
5. Finally, we find the derivative of xln(sin(x)) using the product rule. The derivative is ln(sin(x)) + x * d/dx(ln(sin(x))).