Final answer:
To find the differential of the function y = 6xlnx, we use the product rule of differentiation. The differential dy is equal to 6dx + 6lnx dx.
Step-by-step explanation:
To find the differential of the function y = 6xlnx, we need to use the product rule of differentiation. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:
dy/dx = u(x)v'(x) + u'(x)v(x)
In this case, u(x) = 6x and v(x) = lnx. Taking the derivatives of u(x) and v(x), we have:
u'(x) = 6
v'(x) = 1/x
Now we can use the product rule formula to find the differential dy:
dy = u(x)v'(x)dx + u'(x)v(x)dx
Substituting in the values, we get:
dy = (6x)(1/x)dx + (6)(lnx)dx
Simplifying, we have:
dy = 6dx + 6lnx dx