Find dy/dt.
y=cos^4(pi(t)-20)
If you could please show step-by-step, it would be much appreciated. Thanks in advance!
y = cos^4(pi*t-20)
Note that the derivative of cos(x) = -sin(x) and the derivative of x^n = n*x^(n-1). We'll apply them here, and we'll also use chain rule.
dy/dt
= 4 * pi * cos^3 (pi*t - 20) * (-sin (pi*t - 20))
Hope this helps~ :3
Thanks for the help :)
To find dy/dt, we need to apply the chain rule of differentiation, since we have a composite function.
Let's break it down step-by-step:
Step 1: Identify the inner and outer functions.
In this case, the inner function is pi(t)-20, and the outer function is cos^4(x).
Step 2: Compute the derivative of the outer function.
The derivative of cos^4(x) can be found by applying the chain rule. Let's denote the outer function as u:
u = cos(x)
Then, applying the chain rule, we have:
du/dx = -sin(x)
Now, using the power rule, we differentiate u^4 with respect to u:
d(u^4)/du = 4u^(4-1) = 4u^3
So, the derivative of cos^4(x) with respect to x is:
-4sin(x) * cos^3(x).
Step 3: Compute the derivative of the inner function.
We need to find d(pi(t)-20)/dt. Since pi is a constant, its derivative is 0. So, we only need to consider the -20 term, which has a derivative of 0.
Step 4: Apply the chain rule.
Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function:
dy/dt = dy/d(pi(t)-20) * d(pi(t)-20)/dt
Since d(pi(t)-20)/dt = 0, we can ignore this term.
Step 5: Simplify the result.
dy/dt = -4sin(pi(t)-20) * cos^3(pi(t)-20)
And that's the final answer for dy/dt.