Find the derivative of

4 sec^7 (πt − 6)?

well, we know we keep the 4 :)

Let z = 4(sec u)^7 and u = pi t - 6
then
dz/du = 4 * 7 (sec u)^6 d sec u/du

= 28 (sec u)^6 (sec u )(tan u)

= 28 sec^7 u tan u
dz/dt = dz/du * du/dt = dz/du * pi
so
dz/dt = 28 pi sec^7(pi t-6)tan(pi t-6)

To find the derivative of the function 4 sec^7 (πt − 6), we can use the chain rule. The chain rule states that if we have a composite function f(g(x)), the derivative is given by f'(g(x)) * g'(x).

Let's break down the problem and use the chain rule step-by-step:

Step 1: Identify the outer function and its derivative.
The outer function is sec^7(x), and its derivative is given by d/dx sec^7(x) = 7(sec(x))^6 * sec(x)tan(x).

Step 2: Identify the inner function and its derivative.
The inner function is (πt - 6), and its derivative is d/dx (πt - 6) = π.

Step 3: Apply the chain rule.
Using the chain rule, the derivative of 4 sec^7 (πt - 6) is given by:

d/dt [4 sec^7 (πt - 6)] = 4 * [7(sec(πt - 6))^6 * sec(πt - 6)tan(πt - 6)] * π

Simplifying further, we have:

= 4 * [7(sec(πt - 6))^6 * sec(πt - 6)tan(πt - 6)] * π

Therefore, the derivative of 4 sec^7 (πt - 6) with respect to t is 4 * [7(sec(πt - 6))^6 * sec(πt - 6)tan(πt - 6)] * π.

To find the derivative of the given function, we can use the chain rule.

The chain rule states that if we have a composite function y = f(g(x)), where f(u) is a differentiable function and g(x) is also differentiable, then the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x).

In this case, our function is y = 4 sec^7 (πt − 6). Let's break it down to simplify the process.

Step 1: Rewrite the function
We can rewrite the function as y = 4 * (sec(πt − 6))^7.

Step 2: Identify the inner and outer functions
The inner function is g(x) = πt − 6, and the outer function is f(u) = u^7.

Step 3: Find the derivatives of the inner and outer functions
The derivative of the inner function g'(x) = π (since the derivative of πt is π).
The derivative of the outer function f'(u) = 7u^6 (since the power rule states that the derivative of u^n is n*u^(n-1)).

Step 4: Apply the chain rule
Now, we can apply the chain rule by multiplying the derivatives of the inner and outer functions:

dy/dx = f'(g(x)) * g'(x)
= 7(sec(πt − 6))^6 * π

So, the derivative of 4 sec^7 (πt − 6) with respect to t is 7π(sec(πt − 6))^6.