Differentiate x^(4) + x^(1/2) + 3x^(5)

When considering a term in the form of xⁿ, we use the rule nx^n-1 when differentiating.

So, we multiply the power of x by the coefficient (number in front) of x and then minus one from the power of x. In this example, we take each term by itself and differentiate it separately.

Taking the first term: x⁴

Here, since there is no number in front of x, this means the coefficient of x is 1, so we multiply the power of x by 1 to give us 4, which is our new number in front of the x. The 4 is the n in the nx^n-1.

We then take one away from the power (4-1) to give us 3, our new power of x. This makes 3 out n-1 in the nx^n-1.

So, this leaves us with 4x³, the differential of the first term.

So, now we differentiate the next term: x^1/2, even though the power is a fraction, it still works in the same way.

Bring the 1/2 down in front of the x and multiply it by the current coefficient of x (in this case, it is 1 again). This leaves us with 1/2 in front of the x.

We then take 1 away from the current power of x, so 1/2 -1 = -1/2 leaving us with a new power of -1/2.

Therefore, we put this together and this term differentiates to 1/2x^-1/2

Finally, the last term: 3x⁵. This works in the same way despite the number in front of the x.

Bring the power of x down in front of the x and multiply it by the coefficient of x so 3x5, which leaves us with 15.

Then, minus 1 from the power of x, so 5-1=4, giving us the new power of 4.

Hence the answer for this part is 15x⁴, again fitting the nx^n-1 form.

The full answer therefore, is just each of these terms added together. All our coefficients here are positive but watch out on other questions to write the terms with negative coefficients in as subtractions.

Answer: 4x³+1/2x^-1/2+15x⁴