Find the Derivative - d/dx (sin(x))^( natural log of x)

Problem

d(sin(x))/d(x)

Solution

1. Identify the function as a power-tower form y=ƒ*(x)g(x) which requires logarithmic differentiation or the identity ab=e(b*ln(a))

2. Rewrite the expression using the exponential identity to prepare for the chain rule.

y=e(ln(x)*ln(sin(x)))

3. Apply the chain rule to differentiate the exponential function, which involves taking the derivative of the exponent.

d(y)/d(x)=e(ln(x)*ln(sin(x)))⋅(d(ln(x))*ln(sin(x)))/d(x)

4. Apply the product rule to the derivative of the exponent, where the two functions are u=ln(x) and v=ln(sin(x))

(d(ln(x))*ln(sin(x)))/d(x)=d(ln(x))/d(x)⋅ln(sin(x))+ln(x)⋅d(ln(sin(x)))/d(x)

5. Differentiate the individual components using the basic derivative rules and the chain rule for ln(sin(x))

d(ln(x))/d(x)=1/x

d(ln(sin(x)))/d(x)=1/sin(x)⋅cos(x)=cot(x)

6. Substitute these derivatives back into the product rule expression.

(d(ln(x))*ln(sin(x)))/d(x)=ln(sin(x))/x+ln(x)*cot(x)

7. Combine all parts and substitute the original expression back in for the exponential term.

d(y)/d(x)=sin(x)*(ln(sin(x))/x+ln(x)*cot(x))

Final Answer

d(sin(x))/d(x)=sin(x)*(ln(sin(x))/x+ln(x)*cot(x))

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