Question

Suppose u and v are functions of x that are differentiable at x=0 and that u(0)=1, u'(0) = -2, v(0)= -7 and v'(0) = -5. Find the values of the following expressions at x=0:

d/dx(uv) at x=0

Answer 1

Using the product rule for differentiation, the value of d/dx(uv) at x=0 for the given functions u and v with their respective values and derivatives at x=0 is 9.

Step-by-step explanation:

The student is asking for the derivative of the product of two functions u(x) and v(x) at the point x=0. To find the value of d/dx(uv) at x=0, we can use the product rule for differentiation, which states that the derivative of a product of two functions is given by:

d/dx(uv) = u'v + uv'.

Given the values u(0)=1, u'(0)=-2, v(0)=-7, and v'(0)=-5, we can substitute these into the product rule to find the derivative at x=0:

d/dx(uv) at x=0 = u'(0)v(0) + u(0)v'(0)

= (-2)(-7) + (1)(-5)

= 14 - 5

= 9.

Therefore, the value of d/dx(uv) at x=0 is 9.