Mathematics, five units · Grade 12

How do you differentiate a root and an exponential function?

Rewrite a root as a fractional power and the ordinary power rule applies unchanged — a square root is a power of one half. The exponential function with base e is its own derivative, which is what makes e the natural base. When there is an expression inside either of them, finish with the chain rule.

Learning objectives

A root is a power

The square root of x is x to the power of one half. Once it is written that way there is no new rule: bring the exponent down and reduce it by one, exactly as for any other power.

Reducing one half by one gives minus one half, which is why the derivative of a square root has a root in the denominator. That is not a special formula but the power rule doing what it always does.

The same works for any root. A cube root is a power of one third, and a reciprocal is a negative power — rewriting first turns three apparently separate rules back into one.

The exponential function

The function e to the x differentiates to itself. No coefficient appears and the exponent does not change, which is unique to this base and is the reason e is called natural.

For another base, a coefficient appears — the natural logarithm of that base. That is why five-unit work keeps everything in base e wherever it can.

When there is an expression inside

A root over a whole expression, or e raised to an expression, is a composition. Differentiate the outer function as usual and then multiply by the derivative of what is inside.

The inner derivative is the step people drop, and it is invisible when the inside is just x, because it is one. That is exactly why the habit fails the first time the inside is anything else.

Worked examples

  1. Differentiate the square root of x

    1. Rewrite as x to the power of one half
    2. Bring the exponent down: one half in front
    3. Reduce the exponent by one: minus one half

    Answer: One over twice the square root of x

  2. Differentiate e to the power 3x

    1. The outer function is e to a power, which differentiates to itself
    2. The inside is 3x, whose derivative is 3
    3. Multiply by the inner derivative

    Answer: 3 times e to the power 3x

  3. Differentiate the square root of (x squared + 1)

    1. Rewrite as the bracket to the power of one half
    2. Outer derivative: one half times the bracket to the power minus one half
    3. Inner derivative: 2x
    4. Multiply and simplify

    Answer: x over the square root of (x squared + 1)

Common mistakes

Looking for a special rule for roots
A root is a fractional power, and the power rule covers it. Treating it as a separate case invites a formula misremembered under pressure.
Adding a coefficient when differentiating e to the x
With base e the function is its own derivative — no coefficient and no change of exponent. A coefficient appears only for a different base, and there it is the natural logarithm of that base.
Forgetting the inner derivative
Any expression other than a bare x inside means the chain rule. The step is invisible when the inside is x, which is precisely why it gets dropped everywhere else.

What to remember

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