To find the derivative of the function y = sin(5x)ln(x), we will use logarithmic differentiation. This technique is especially useful when the function is a product of multiple functions or involves variable exponents.
First, we take the natural logarithm of both sides:
ln(y) = ln(sin(5x)) + ln(ln(x))Next, we differentiate both sides with respect to x:
rac{1}{y} rac{dy}{dx} = rac{1}{sin(5x)} rac{d}{dx}(sin(5x)) + rac{1}{ln(x)} rac{d}{dx}(ln(x))Now we compute the derivatives:
- The derivative of sin(5x) is 5cos(5x).
- The derivative of ln(x) is rac{1}{x}.
Thus, we can rewrite our equation:
rac{1}{y} rac{dy}{dx} = rac{5cos(5x)}{sin(5x)} + rac{1}{ln(x)} rac{1}{x}Now, combine the fractions on the right:
rac{1}{y} rac{dy}{dx} = 5cot(5x) + rac{1}{x ln(x)}Next, we multiply through by y to isolate rac{dy}{dx}:
rac{dy}{dx} = y(5cot(5x) + rac{1}{x ln(x)})Finally, substitute back the original function for y:
rac{dy}{dx} = sin(5x)ln(x)(5cot(5x) + rac{1}{x ln(x)})This is the derivative of the function y = sin(5x)ln(x).