Question

Efine the derivative and anti-derivative of a function. Find, from the first principle, the derivative of sin(ax+b).

A. The derivative is the slope of the tangent line; anti-derivative is the area under the curve.
B. The derivative is the area under the curve; anti-derivative is the slope of the tangent line.
C. The derivative is the rate of change; anti-derivative is the reverse process of differentiation.
D. The derivative is the reverse process of differentiation; anti-derivative is the rate of change.

Answer 1

The derivative is the rate of change of a function, and the anti-derivative is the reverse process of differentiation. using first principles, we derive that the derivative of sin(ax + b) is a*cos(ax + b). this reinforces the understanding that derivatives have significant dimensional properties.

Step-by-step explanation:

The derivative of a function represents the rate of change of the function with respect to a variable, often time or space. In contrast, an anti-derivative, also known as an integral, represents the reverse process of differentiation, essentially describing the accumulation of quantities such as area under a curve.

To find the derivative of sin(ax + b) from the first principle, we start by using the definition of the derivative as the limit of the difference quotient:

1. Start with the function f(x) = sin(ax + b).
2. Consider the difference quotient: (f(x + h) - f(x)) / h where h approaches 0.
3. Substitute and apply the trigonometric limit: lim(h->0) [(sin(a(x + h) + b) - sin(ax + b)) / h].
4. Use trigonometric identities to simplify the expression: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
5. Simplify the limit to obtain the derivative.

This process results in the derivative of sin(ax + b) being a*cos(ax + b), reflecting the rate of change of the sine function with respect to x. the physical significance of these operations is reinforced through their dimensional properties, confirming that for variables v and t, the dimension of the derivative of v with respect to t is the ratio of their dimensions.