Use the Chain Rule to find ∂z/∂s and ∂z/∂t.
z = tan(u/v), u = 7s + 4t, v = 4s − 7t
thank you!!!!!!
Oh boy, it's time to dive into some derivatives! Let's get started.
To find ∂z/∂s, we need to use the chain rule. So let's break it down step by step.
First, let's differentiate z with respect to u:
∂z/∂u = sec^2(u/v)
Then, let's differentiate u with respect to s:
∂u/∂s = 7
Next, let's differentiate v with respect to s:
∂v/∂s = 4
Now we can use the chain rule to find ∂z/∂s:
∂z/∂s = (∂z/∂u) * (∂u/∂s)
= sec^2(u/v) * 7
Alrighty, now let's move on to ∂z/∂t. We'll follow a similar process.
First, let's differentiate z with respect to u again:
∂z/∂u = sec^2(u/v)
Then, let's differentiate u with respect to t:
∂u/∂t = 4
Next, let's differentiate v with respect to t:
∂v/∂t = -7
Now we can use the chain rule to find ∂z/∂t:
∂z/∂t = (∂z/∂u) * (∂u/∂t)
= sec^2(u/v) * 4
So there you have it! ∂z/∂s = 7sec^2(u/v) and ∂z/∂t = 4sec^2(u/v). Happy derivatives!
To find ∂z/∂s and ∂z/∂t using the Chain Rule, we need to differentiate the function z = tan(u/v) with respect to s and t.
First, let's find ∂z/∂s:
Step 1: Find ∂z/∂u (partial derivative of z with respect to u)
To find ∂z/∂u, we need to use the chain rule. The chain rule states that if z = f(g(u)), then ∂z/∂u = f'(g(u)) * g'(u).
In this case, f(u) = tan(u/v) and g(u) = u. So, ∂z/∂u = f'(g(u)) * g'(u).
Let's find f'(u):
f(u) = tan(u/v)
To find the derivative of f(u), we can rewrite it as f(u) = tan(u) * (1/v). Now, we can find the derivative of this expression with respect to u using trigonometric identities.
f'(u) = sec^2(u)*(1/v)
Next, let's find g'(u):
g(u) = u
g'(u) = 1
Now, we can substitute the derivatives we found back into the chain rule equation:
∂z/∂u = f'(g(u)) * g'(u)
∂z/∂u = sec^2(u)*(1/v) * 1
∂z/∂u = sec^2(u)/v
Step 2: Find ∂u/∂s (partial derivative of u with respect to s)
Given that u = 7s + 4t, we can differentiate u with respect to s to find ∂u/∂s:
∂u/∂s = 7
Now, we can find ∂z/∂s by using the chain rule:
∂z/∂s = (∂z/∂u) * (∂u/∂s)
∂z/∂s = (sec^2(u)/v) * 7
Step 3: Find ∂z/∂t (partial derivative of z with respect to t)
Similarly, we can find ∂z/∂t using the chain rule:
∂z/∂t = (∂z/∂u) * (∂u/∂t)
To find ∂u/∂t, we can differentiate u with respect to t:
∂u/∂t = 4
Now, substitute the derivatives we found back into the equation:
∂z/∂t = (sec^2(u)/v) * 4
Therefore, ∂z/∂s = (sec^2(u)/v) * 7 and ∂z/∂t = (sec^2(u)/v) * 4.
The answer is not too complicated. If you go to wolframalpha.com and enter
d/ds (tan((7s+4t)/(4s-7t))
and then click the "show steps" button, you can see the steps taken.