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

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To determine the differentiability of the function 3g at t = 7, we need to understand the concept of differentiability for a function. A function is differentiable at a point if it has a derivative at that point.

Given that g(t) is differentiable at t = 7, it means that the derivative g'(7) exists.

The function 3g(t) is a scalar multiple of g(t). The derivative of a scalar multiple of a function is the scalar multiplied by the derivative of the function. This is a basic rule of differentiation.

Therefore, the derivative of 3g(t) with respect to t is 3 times the derivative of g(t). Mathematically, this can be expressed as (3g)'(t) = 3 * g'(t).

Since g'(7) exists, it follows that (3g)'(7) = 3 * g'(7) also exists. Thus, the function 3g is differentiable at t = 7.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Differentiability

Differentiability of a function at a point means that the function has a defined derivative at that point. This implies the function is smooth and continuous at that point, without any sharp corners or discontinuities. For a function g(t) to be differentiable at t = 7, it must have a well-defined tangent line at that point.

Derivative of a Constant Multiple

The derivative of a constant multiple of a function is the constant multiplied by the derivative of the function. Mathematically, if g(t) is differentiable, then the derivative of 3g(t) is 3 times the derivative of g(t). This rule ensures that scaling a function by a constant does not affect its differentiability.

Linearity of Differentiation

Differentiation is a linear operation, meaning it respects addition and scalar multiplication. This property implies that if g(t) is differentiable at a point, then any linear transformation of g(t), such as 3g(t), is also differentiable at that point. Thus, knowing g(t) is differentiable at t = 7 ensures 3g(t) is differentiable at t = 7.