Question

Use logarithmic differentiation to find the derivative of the function. y = (sin(5x))ln(x).

a) dy/dx=5cos(5x)ln(x)+sin(5x)/x
b) dy/dx=cos(5x)ln(x)+5sin(5x)/x
c) dy/dx=5cos(5x)ln(x)−sin(5x)/x
d) dy/dx=cos(5x)ln(x)−5sin(5x)/x

Answer 1

To find the derivative of the function y = (sin(5x))ln(x) using logarithmic differentiation, we take the natural logarithm of both sides, simplify using logarithmic properties, apply implicit differentiation, and substitute the original function back in. The derivative is 5cos(5x)ln(x) + sin(5x)/x.

Step-by-step explanation:

To find the derivative of the function y = (sin(5x))ln(x) using logarithmic differentiation, we will take the natural logarithm of both sides and apply the logarithmic differentiation rules.

Step 1: Take the natural logarithm of both sides of the equation:
ln(y) = ln((sin(5x))ln(x))

Step 2: Simplify using logarithmic properties:
ln(y) = ln(sin(5x)) + ln(ln(x))

Step 3: Differentiate implicitly with respect to x:
(1/y)*(dy/dx) = (1/sin(5x))*(5cos(5x)) + (1/ln(x))*(1/x)

Step 4: Solve for (dy/dx):
dy/dx = (y/sin(5x))*(5cos(5x)) + (y/ln(x))*(1/x)

Step 5: Substitute the original function back in:
dy/dx = (sin(5x)ln(x)/sin(5x))*(5cos(5x)) + (sin(5x)ln(x)/ln(x))*(1/x)

Simplifying further, we get:
dy/dx = 5cos(5x)ln(x) + sin(5x)/x

Therefore, the correct answer is option a) dy/dx = 5cos(5x)ln(x) + sin(5x)/x.