Let's solve each part of the problem step by step:

Part (a)

We are given the function: $y = (7t^5 - 8)^6$

• Inner function: The expression inside the power is $7t^5 - 8$. So, let $u = 7t^5 - 8$.

• Outer function: Now, $y$ becomes a power function of $u$, i.e. $y = u^6$

Thus, for part (a):

• $u = 7t^5 - 8$
• $y = u^6$

Part (b)

We are given the function: $P = 12e^{-0.3t}$

• Inner function: The exponent part is $-0.3t$. So, let $u = -0.3t$.

• Outer function: $P$ becomes an exponential function with base $e$, i.e. $P = 12e^u$

Thus, for part (b):

• $u = -0.3t$
• $P = 12e^u$

Part (c)

We are given the function: $C = 11\ln(q^4 + 6)$

• Inner function: The expression inside the logarithm is $q^4 + 6$. So, let $u = q^4 + 6$.

• Outer function: $C$ becomes a logarithmic function, i.e. $C = 11\ln(u)$

Thus, for part (c):

• $u = q^4 + 6$
• $C = 11\ln(u)$

Would you like further details or have any questions?

Relative Questions:

1. What is the general process for expressing a function as a composition of two functions?
2. How can we differentiate a composite function using the chain rule?
3. How would the outer function change if $y = (7t^5 - 8)^7$ instead?
4. Can you express $P = 12e^{0.3t}$ as a composition of functions, and what would change?
5. How does the choice of inner function affect the derivative of composite functions?

Tip:

When identifying composite functions, always look for the "outer" function (e.g., a power, exponential, or logarithmic function) and the "inner" function (typically what's inside parentheses, exponents, etc.).