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Chapter 2: Problem 2

Find the derivative of \(f(x)=\frac{1}{x^{17}}\). \(f^{\prime}(x)=\) __________

Short Answer

Expert verified
f^{\prime}(x) = -17 x^{-18}

Step by step solution

01

Rewrite the function using exponent rules

Rewrite the given function in a more convenient form for differentiation. The function is given as:\[ f(x) = \frac{1}{x^{17}} \]This can be rewritten using the exponent rule \( \frac{1}{x^n} = x^{-n} \):\[ f(x) = x^{-17} \]
02

Apply the power rule

To differentiate, apply the power rule \( \frac{d}{dx}(x^n) = n x^{n-1} \). Here, we need to find the derivative of \( f(x) = x^{-17} \):\[ f^{\prime}(x) = \frac{d}{dx}(x^{-17}) = -17 x^{-18} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is a fundamental tool for differentiating functions. It states that if you have a function of the form \(f(x) = x^n\), its derivative is given by \(f'(x) = n x^{n-1}\). This rule makes it simple to find the derivative of polynomial expressions.
For example, if \(f(x) = x^3\), according to the power rule, the derivative is \(f'(x) = 3x^2\).

To apply the power rule, you just:
  • Multiply by the exponent (the number that x is raised to).
  • Reduce the exponent by one.
It's a straightforward two-step process that applies to any power of x, positive or negative.
Differentiation
Differentiation is a key concept in calculus that deals with finding the rate at which a function changes at any given point.
Essentially, it is used to calculate the slope of a function at a specific point, which is known as the derivative.

The derivative of a function \(f(x)\) is denoted as \(f'(x)\) and is defined as:
\(f'(x) = \frac{d}{dx} f(x) = \frac{f(x + h) - f(x)}{h} \text{ as } h \rightarrow 0\).

To differentiate more complex functions, you often use rules and techniques like:
  • The Power Rule
  • Product Rule
  • Quotient Rule
  • Chain Rule

In our example, we used differentiation to find how the function \(f(x) = \frac{1}{x^{17}} \) changes with respect to x.
Exponent Rules
Exponent rules are crucial for rewriting functions in a form suitable for differentiation.
They are a set of guidelines that help simplify expressions involving exponents.

Key exponent rules include:
  • \(a^m \times a^n = a^{m+n}\)
  • \(\frac{a^m}{a^n} = a^{m-n}\)
  • \((a^m)^n = a^{mn}\)
  • \(a^{-n} = \frac{1}{a^n}\)

For our exercise, we used the last rule to transform \( \frac{1}{x^{17}} \) into \( x^{-17} \). This simplification made it easy to apply the power rule for differentiation.
Understanding and mastering these rules can significantly simplify your work with algebraic and calculus problems.

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Most popular questions from this chapter

Find an equation for the line tangent to the graph of \(f\) at \((2,27),\) where \(f\) is given by \(f(x)=4 x^{3}-4 x^{2}+11\). \(y=\) _________

Find the slope of the tangent to the curve \(x^{3}+2 x y+y^{2}=64\) at (1,7). The slope is \(\square\). (Enter undef if the slope is not defined at this point.)

Consider the function \(g(x)=x^{2 x}\), which is defined for all \(x>0\). Observe that \(\lim _{x \rightarrow 0^{+}} g(x)\) is indeterminate due to its form of \(0^{0}\). (Think about how we know that \(0^{k}=0\) for all \(k>0\), while \(b^{0}=1\) for all \(b \neq 0\), but that neither rule can apply to \(0^{0}\).) a. Let \(h(x)=\ln (g(x))\). Explain why \(h(x)=2 x \ln (x)\). b. Next, explain why it is equivalent to write \(h(x)=\frac{2 \ln (x)}{\frac{1}{x}}\). c. Use L'H么pital's Rule and your work in (b) to compute \(\lim _{x \rightarrow 0^{+}} h(x)\). d. Based on the value of \(\lim _{x \rightarrow 0^{+}} h(x)\), determine \(\lim _{x \rightarrow 0^{+}} g(x)\).

Compute the following limits using l'H么pital's rule if appropriate. Use INF to denote \(\infty\) and MINF to denote \(-\infty\). \(\lim _{x \rightarrow 0} \frac{1-\cos (7 x)}{1-\cos (6 x)}=\) ___________ \(\lim _{x \rightarrow 1} \frac{7^{x}-6^{x}-1}{x^{2}-1}=\) __________

For the curve given by the equation \(\sin (x+y)+\cos (x-y)=1,\) find the equation of the tangent line to the curve at the point \(\left(\frac{\pi}{2}, \frac{\pi}{2}\right)\).
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