Derivative of e^x + x^2 Step by Step

Step by step

Question: find the derivative of f(x) = eˣ + x².

f(x) = e^x + x²

eˣ is special: its derivative is itself.

Its slope always equals its height.

e^x → e^x

And x to a power: the power rule, as before.

xⁿ → nx^{n − 1}

Use the rule on each term.

f′(x) = e^x + 2x^(2 − 1)

Careful!

The power rule is only for x to a number.

In e to the x, the x is the power.

So e to the x stays e to the x.

Careful: not e^x → xe^{x − 1} ✗

So f′(x) = eˣ + 2x.

= e^x + 2x

Let's check: the derivative is the slope.

At x = 1, the rule gives 4.7183, with a calculator in radians.

f′(1) = e + 2 ≈ 4.7183

Now the slope between two very close points: x = 1 and x = 1.001.

A calculator gives about 4.72.

Very close to 4.7183.

Right!

f(1.001) − f(1)0.001 ≈ 4.72

Let's look back.

First: the rules.

Then: each term.

Then: tidy up.

Last: check.

Answer

= e^x + 2x