Question

Find the differential of each function.
a) y = x^2 sin(6x)
b) y = ln(sqrt(3 + t^2))

Answer 1

To find the differential of the function y = x^2 sin(6x), first find the derivative of x^2 and apply the chain rule to sin(6x). For the function y = ln(sqrt(3 + t^2)), apply the chain rule to the inner function and then apply the derivative of ln(u).

Step-by-step explanation:

Differential of the function y = x^2 sin(6x)

Step 1: Find the derivative of x^2 with respect to x.

dy/dx = 2x

Step 2: Apply the chain rule to sin(6x) by differentiating the outer function and multiplying it by the derivative of the inner function.

dy/dx = 6x cos(6x)

So, the differential of y = x^2 sin(6x) is dy/dx = 2x + 6x cos(6x).

Differential of the function y = ln(sqrt(3 + t^2))

Step 1: Apply the chain rule to the inner function sqrt(3 + t^2) by differentiating the outer function and multiplying it by the derivative of the inner function.

dy/dt = 1 / (2 sqrt(3 + t^2)) * (2t)

Step 2: Apply the derivative of ln(u) where u = sqrt(3 + t^2).

dy/dt = 1 / (2 sqrt(3 + t^2)) * (2t) = t / sqrt(3 + t^2)

So, the differential of y = ln(sqrt(3 + t^2)) is dy/dt = t / sqrt(3 + t^2).