Question

Compute the higher derivative. $$ \frac{d^{3}}{d x^{3}}(9-x)^{8} $$

   Compute the higher derivative.
$$
\frac{d^{3}}{d x^{3}}(9-x)^{8}
$$

Calculus for AP

Calculus for AP

Jon Rogawski & Colin… 3rd Edition

Chapter 3, Problem 75

Instant Answer

Solved by Expert Linh Vu

10/17/2020

Step 1

We can use the power rule for differentiation, which states that the derivative of $x^n$ is $nx^{n-1}$. However, since our function is $(9-x)^8$, we need to use the chain rule as well. The chain rule states that the derivative of a composite function is the Show more…

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Compute the higher derivative. $$ \frac{d^{3}}{d x^{3}}(9-x)^{8} $$

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Key Concepts

Chain Rule

The chain rule is a fundamental technique for differentiating composite functions, where one function is nested within another. In this problem, the function (9-x)^8 is composed of an outer power function and an inner linear function, making the chain rule critical for appropriately computing the derivative by differentiating the outer function and then multiplying by the derivative of the inner function.

Power Rule

The power rule is a basic rule for differentiation that states if f(x) = x^n, then its derivative is f'(x) = n*x^(n-1). In this problem, the power rule is applied to the function (9-x)^8 during each differentiation step to reduce the exponent and compute the derivatives efficiently.

Higher Order Derivatives

Higher order derivatives involve taking the derivative of a function more than once. In the context of this problem, calculating the third derivative requires applying differentiation rules sequentially. This concept emphasizes understanding how a function's rate of change itself changes and is essential for problems in calculus that involve motion, curvature, or Taylor series expansions.

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