Derivative Of A Function

Explanation
The derivative of f(x) = x^4 is calculated by applying the power rule. The power rule states that if f(x) = x^n, then the derivative of f(x) is f'(x) = nx^(n-1). In this case, n = 4, so the derivative of f(x) = x^4 is f'(x) = 4x^(4-1) = 4x^3. Therefore, Option 3 (4x^3) and Option 4 (4x^3) represent the correct derivatives of f(x) = x^4.

Explanation
The graph of G(x) shows that the value of G(x) is constant at 0.2 for all values of x. Therefore, the derivative G'(x) is equal to 0 for all values of x, including x = 7. Hence, the estimate for G'(7) is 0.

Explanation
The derivative of a function does not exist at points where there is a sharp change in the slope of the graph or where the graph has a vertical tangent line. In this case, looking at the given values of x, we can see that x = -2, x = 0, x = 2, and x = 3 are the points where the derivative of the graphed function does not exist. At these points, there are either sharp changes in slope or vertical tangent lines.

Explanation
The statement "All continuous functions are differentiable" is false. While it is true that all differentiable functions are continuous, the reverse is not always true. There are continuous functions that are not differentiable at certain points or on certain intervals. A classic example is the absolute value function, which is continuous but not differentiable at x = 0. Therefore, the correct answer is false.

Explanation
This statement is true because all differentiable functions are continuous. Differentiability implies continuity, meaning that if a function is differentiable at a point, it must also be continuous at that point. This is because the definition of differentiability includes the requirement that the function has a well-defined derivative at every point, and having a derivative implies that the function is continuous. Therefore, all differentiable functions are continuous.

Explanation
If a function has a discontinuity at x = a, it means that the function is not continuous at that point. The derivative of a function measures the rate of change of the function at a particular point. Since a function is not continuous at x = a, it means that there is a sudden jump or break in the function's values at that point. This sudden jump or break makes it impossible to calculate the derivative at x = a, as the concept of rate of change does not apply when there is a discontinuity. Therefore, the statement that the function does not have a derivative at x = a is true.

Explanation
The derivative does not exist at a cusp because at a cusp, the function has a sharp point where the tangent line cannot be defined. On the other hand, the derivative does exist at a corner because at a corner, the function has a well-defined tangent line on either side of the corner. Therefore, the statement that the derivative does not exist at a cusp but does exist at a corner is false.

Explanation
The statement is true because when applying algebraic techniques to find the derivative of a function from the definition of derivative, one uses the limit of a difference quotient. This involves taking the limit as the difference in x-values approaches zero, which gives the instantaneous rate of change of the function at a specific point. Therefore, the statement is correct.