Question

From what you know about the graphs of functions (i) through (v), pick out the one that is its own derivative. i. \(y=\sin x \quad\) ii. \(y=x \quad\) iii. \(y=\sqrt{x}\) iv. \(y=e^{x} \quad\) v. \(y=x^{2}\)

Short answer

The function that is its own derivative among the given options is \(y=e^{x}\)

Step-by-step solution

Find derivative of \(y=\sin x\)

Derivative of \(y=\sin x\) is \(y'=\cos x\). This does not equal to the original function \(y=\sin x\) so it can't be the answer.

Find derivative of \(y=x\)

Derivative of \(y=x\) is \(y'=1\). This also does not equal to the original function \(y=x\) so it can't be the answer.

Find derivative of \(y=\sqrt{x}\)

Derivative of \(y=\sqrt{x}\) is \(y' = \frac{1}{2\sqrt{x}}\). This also does not equal the original function \(y=\sqrt{x}\) hence it also can't be the answer.

Find derivative of \(y=e^{x}\)

Derivative of \(y=e^{x}\) is \(y'=e^{x}\). This equals to the original function \(y=e^{x}\). Therefore, \(y=e^{x}\) is the function that is its own derivative. Since we've found the solution, there's no need to compute the derivative of \(y=x^{2}\).

Final confirmation

Confirm that \(y'=e^{x}\) is equal to the original function \(y=e^{x}\).