Question

What functions have derivative equal to x^7?

Answer 1

The function with a derivative equal to x^7 is f(x) = (1/8)x^8 + C, where C is the constant of integration.

Explanation:

To find a function whose derivative is x^7, we can use the reverse power rule of differentiation. The reverse power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).

Using this rule, we can see that if f(x) = (1/8)x^8, then f'(x) = 8(1/8)x^(8-1) = x^7. However, we also need to include the constant of integration, which we'll call C.

So the function with a derivative of x^7 is f(x) = (1/8)x^8 + C.

To check that this is correct, we can take the derivative of f(x) using the power rule of differentiation:

f'(x) = d/dx [(1/8)x^8 + C]

= (1/8)d/dx (x^8) + d/dx (C)

= (1/8)(8x^7) + 0

= x^7

So we can see that f(x) = (1/8)x^8 + C is a valid function whose derivative is x^7

Answer 2

To find functions whose derivative is equal to \(x^7\), we can use the process of integration. Since the derivative of a function is the rate of change of that function, integrating a function will "undo" the differentiation process and give us the original function.

The antiderivative (or indefinite integral) of \(x^7\) can be found by adding 1 to the exponent and dividing by the new exponent:

\[\int x^7 \, dx = \frac{1}{8}x^8 + C\]

Where C is the constant of integration.

Thus, the functions whose derivative is equal to \(x^7\) are of the form:

\[f(x) = \frac{1}{8}x^8 + C\]

Where C is any constant.