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Calculate the following and give your answer to 0 decimal places accurately in the space below: (don't include any units)

Let's break down the problem step-by-step. The given function is:

\[
h(t) = 2t^4 + 5x^3 - 6t - 7x
\]

You are asked to find the derivative \( h'(t) \) and evaluate it at \( t = 4 \). Note that only terms with \( t \) are relevant here since we are taking the derivative with respect to \( t \).

1. Differentiate each term with respect to \( t \):

- The derivative of \( 2t^4 \) is \( 8t^3 \).
- The derivative of \( 5x^3 \) (which is a term with \( x \), not \( t \)) is treated as a constant, so its derivative is 0.
- The derivative of \( -6t \) is \( -6 \).
- The derivative of \( -7x \) (again, involving \( x \)) is 0.

So, the derivative \( h'(t) \) is:

\[
h'(t) = 8t^3 - 6
\]

2. Now, evaluate \( h'(t) \) at \( t = 4 \):

\[
h'(4) = 8(4^3) - 6 = 8(64) - 6 = 512 - 6 = 506
\]

Therefore, the answer is:

\[
h'(4) = 506
\]

Would you like more details on the steps? Here are some related questions to further your understanding:

1. How do we treat the terms involving \( x \) when differentiating with respect to \( t \)?
2. What are the general rules of differentiation for power functions?
3. Why do constant terms disappear when taking the derivative?
4. How do you find the second derivative of a function like this?
5. What is the importance of evaluating derivatives at specific points?

**Tip:** Always be mindful of which variable you're differentiating with respect to, especially when a function involves multiple variables.

Calculate the following derivative by making use of the differentiation rule(s): h(t) = 2t^4 + 5x^3 - 6t - 7x. Calculate h'(4).

Learn how to differentiate h(t) = 2t^4 + 5x^3 - 6t - 7x and evaluate the derivative at t = 4. This calculus problem covers differentiation concepts such as the power rule. Perfect for Grades 11-12 or college calculus students.

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