Question

Given y = f(u) and u = g(x), find dy/dx using Leibniz's notation for the chain rule: dy/dx = dy/du * du/dx. y = 5u³, u = 3sqrt(x).

a) 15sqrt(x)
b) 15xsqrt(x)
c) 15sqrt3
d) 15sqrt(x³)

Answer 1

To find dy/dx using Leibniz's notation for the chain rule, we need to find dy/du and du/dx. Given that y = 5u³ and u = 3sqrt(x), we can express them as y = 5u³ and u = 3sqrt(x). First, find dy/du as 15u². Next, find du/dx as 3/(2sqrt(x)). Finally, substitute the values into the chain rule formula to get dy/dx = 45u²/(2sqrt(x)).

Step-by-step explanation:

To find dy/dx using Leibniz's notation for the chain rule, we need to find dy/du and du/dx. Given that y = f(u) and u = g(x), we can express them as y = 5u³ and u = 3√x. Let's calculate the derivatives:

First, find dy/du. Differentiating y = 5u³ with respect to u, we get dy/du = 15u².

Next, find du/dx. Differentiating u = 3√x with respect to x, we get du/dx = 3/(2√x).

Finally, substitute the values into the chain rule formula: dy/dx = (dy/du) * (du/dx). Substituting, we have dy/dx = (15u²) * (3/(2√x)). Simplifying, we get dy/dx = 45u²/(2√x).