Find the differential of each function.
a) e ^ tan(pi*t)
b) sqrt (8+ln(z))
a) Well, the differential of e^tan(pi*t) is like trying to catch a slippery fish with banana peels. So, let's get our clown gloves on and take a dive into the wacky world of differentials!
To find the differential, we'll use a little trick called the chain rule. Remember, clown rules apply here, so buckle up!
Let's start by breaking down the function. We have e^tan(pi*t), so let's call the inside of the exponent u = tan(pi*t). Now, we need to find du.
To find du, we'll use the chain rule. The derivative of u = tan(pi*t) is du = sec^2(pi*t) * (pi*dt). Don't worry, I know it sounds like alphabet soup, but bear with me!
Now, let's focus on the outer function e^u. The differential of e^u is simply e^u * du. So, putting it all together, the differential of e^tan(pi*t) is e^tan(pi*t) * sec^2(pi*t) * (pi*dt).
There you have it! The differential of e^tan(pi*t) is e^tan(pi*t) * sec^2(pi*t) * (pi*dt). Don't worry, I won't charge you any extra for all these clown math shenanigans!
b) Ah, sqrt(8+ln(z)), you certainly know how to make things interesting! I hope you're ready for some clown magic, because we're about to find its differential.
To find the differential, I need to put on my clown spectacles and summon the chain rule once again! Let's break down the function and get those gears turning.
We have sqrt(8+ln(z)), so let's call the inside of the square root u = 8 + ln(z). Now, we need to find du.
Finding du is all about applying the chain rule. The derivative of u = 8 + ln(z) is du = (1/z) * dz.
Now, let's focus on the outer function sqrt(u). The differential of sqrt(u) is (1/2) * sqrt(u) * du. So, putting it all together, the differential of sqrt(8+ln(z)) is (1/2) * sqrt(8+ln(z)) * (1/z) * dz.
Voila! The differential of sqrt(8+ln(z)) is (1/2) * sqrt(8+ln(z)) * (1/z) * dz. I hope my clown math skills didn't make your head spin too much!
To find the differential of each function, we need to differentiate them with respect to the appropriate variables. Let's go through each function step by step.
a) e ^ tan(pi*t):
To find the differential of this function, we'll need to use the chain rule since we have a composition of functions. The chain rule states that if we have a composite function f(g(x)), its derivative can be found by taking the derivative of the outside function f'(g(x)) multiplied by the derivative of the inside function g'(x).
Let's break down the function e ^ tan(pi*t) into its composite functions:
f(x) = e^x (let x = tan(pi*t))
g(x) = tan(pi*t)
Using the chain rule, we'll differentiate f(g(x)):
f'(g(x)) = d/dx(e^x) = e^x
Now, we need to find the derivative of the inside function g(x) = tan(pi*t):
g'(x) = d/dx(tan(pi*t))
To find the derivative of the tangent function, we'll use the chain rule again:
g'(x) = d/dx(tan(pi*t)) = derivative of tan(u) * derivative of (pi*t) with respect to x
= sec^2(pi*t) * (pi)
Now, we can put it all together to find the differential:
d/dt(e ^ tan(pi*t)) = e^tan(pi*t) * sec^2(pi*t) * pi
b) sqrt(8 + ln(z)):
Similar to the previous example, we need to use the chain rule to find the differential of this function. Let's break it down into its composite functions:
f(x) = sqrt(x) (let x = 8 + ln(z))
g(x) = 8 + ln(z)
Differentiating f(g(x)) using the chain rule:
f'(g(x)) = d/dx(sqrt(x)) = 1 / (2 * sqrt(x))
Now, we need to find the derivative of the inside function g(x) = 8 + ln(z):
g'(x) = d/dx(8 + ln(z))
The derivative of a constant is zero, so the derivative of 8 is 0. For the derivative of ln(z), we have:
g'(x) = d/dx(8 + ln(z)) = 0 + 1 / z = 1 / z
Now, we can find the differential:
d/dz(sqrt(8+ln(z))) = 1 / (2 * sqrt(8 + ln(z))) * (1 / z)
So, the differentials of the given functions are:
a) d/dt(e ^ tan(pi*t)) = e^tan(pi*t) * sec^2(pi*t) * pi
b) d/dz(sqrt(8+ln(z))) = 1 / (2 * sqrt(8 + ln(z))) * (1 / z)
a. f(t) = e^[tan (pi *t)]
Let u = tan (pi t)
f(u) = e^u
df/dt = df/du du/dt
du/dt = pi * sec^2 (pi t)
df/du = e^u
df/dt = pi* e^[(tan (pi*t)]*sec^2 (pi*t)
b) Let u = 8 + ln z
f(z) = f[u(z)]) = sqrt u
Use the chain rule, as I did in (a)
df/dz = df/du*du/dz