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Chapter 3: Problem 56

Derivative of multiples Does knowing that a function \(g(t)\) is differentiable at \(t=7\) tell you anything about the differentiability of the function 3\(g\) at \(t=7 ?\) Give reasons for your answer.

Short Answer

Expert verified
Yes, 3g is differentiable at t=7 because g is differentiable.

Step by step solution

01

Understanding Differentiability of g(t)

When a function \(g(t)\) is differentiable at a point \(t=7\), it means that \(g'(7)\) exists. This indicates that there is a well-defined tangent to the curve at \(t=7\). Differentiability also implies continuity at that point.
02

Understanding Function Multiplication by a Constant

When a function \(g(t)\) is multiplied by a constant, say 3, the resulting function is 3\(g(t)\). The basic rule of differentiation states that the derivative of a constant multiplied by a function \(g(t)\) is the constant multiplied by the derivative of the function \(g(t)\). In mathematical terms, if \(g(t)\) is differentiable and has a derivative \(g'(t)\), then \(\frac{d}{dt}(3g(t)) = 3g'(t)\).
03

Applying Differentiation to 3g(t)

Given that \(g(t)\) is differentiable at \(t=7\), it follows that \(g'(7)\) exists. Thus, applying the rule mentioned above, 3\(g'(7)\) also exists, indicating that 3\(g(t)\) is differentiable at \(t=7\). Therefore, the differentiability of \(g(t)\) is directly transferred to 3\(g(t)\) through multiplication by a constant.
04

Conclusion of Derivative Existence

Since \(g(t)\) is differentiable at \(t=7\), the function 3\(g(t)\) is also differentiable at \(t=7\). This is because differentiation is linear in nature, and multiplying by a constant retains differentiability.

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Key Concepts

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

Differentiation Rules
Differentiation rules are fundamental tools used in calculus to determine the derivative of a function. One important rule is the constant multiple rule which simplifies the differentiation process.

This rule states that if you have a constant multiplied by a function, the derivative of this new function will be the constant multiplied by the derivative of the original function. Mathematically, if a function is represented as \(3g(t)\), where 3 is a constant, and \(g(t)\) is differentiable, then the derivative of \(3g(t)\) is \(3g'(t)\).

In our original exercise, understanding that \(g(t)\) is differentiable at a certain point immediately informs us about the differentiability of \(3g(t)\). This is because direct application of the constant multiple rule guarantees that multiplying the function by a constant like 3 does not affect its differentiability. Every constant behaves predictably under differentiation, which is why rules like these are so useful in simplifying complex calculus problems.
Continuity of Functions
Continuity of functions is an important concept closely linked to differentiability. If a function is differentiable at a certain point, it is guaranteed to be continuous at that same point.

Continuity means that you can draw the function's graph without lifting your pen at a specific point 鈥� there are no breaks, jumps, or holes at that place. This is an essential prerequisite of differentiability because it ensures a smooth path for the derivative to follow.
  • When \(g(t)\) is differentiable at \(t=7\), it means \(g(t)\) has no interruptions at this point.
  • Similarly, when multiplied by a constant (as with \(3g(t)\)), the function remains continuous.
This feature is key in differentiating functions since assessing whether a function is continuous can often simplify the examination of its differentiability. For practical purposes, always check for continuity before checking for differentiability.
Linear Properties of Differentiation
Linear properties of differentiation emphasize the linear nature of derivative operations. This means that differentiation behaves predictably under addition and scalar multiplication.

If \(g(t)\) is differentiable at a point, then any constant multiplied by \(g(t)\) retains this differentiable property, due to its linearity. The scaling and addition processes do not disrupt the differentiability at a given point.
  • Scalar multiplication is straightforward: The derivative of any constant times a function is simply the constant times the function's derivative.
  • This is captured in the mathematical expression \(\frac{d}{dt}(3g(t)) = 3g'(t)\), revealing how smooth and predictable differentiation operations are.
In essence, the linear properties ensure that basic arithmetic operations with differentiable functions, such as multiplying by constants or adding functions, will yield results that maintain the integrity of the function's differentiability. All these reinforce the utility and reliability of differentiation rules when handling functions.

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Most popular questions from this chapter

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