Exponential, Logarithmic & Combined Derivatives

You’ll be able to

Differentiate exponential and logarithmic functions, including base-a forms
Apply the chain rule to exponential and logarithmic composites
Combine multiple differentiation rules in a single problem

The natural exponential is its own derivative

The function eˣ is remarkable: d/dx[eˣ] = eˣ. It is the only function (up to a constant multiple) that equals its own derivative, which is why e is the "natural" base. With the chain rule, d/dx[e^(u)] = e^(u)·u′ — the exponential is unchanged and you multiply by the derivative of the exponent.

The natural logarithm and other bases

The natural log has the tidy derivative d/dx[ln x] = 1/x, and with the chain rule d/dx[ln u] = u′/u. For other bases: d/dx[aˣ] = aˣ·ln(a) and d/dx[log_a x] = 1/(x·ln a). The extra factor of ln(a) appears whenever the base is not e — set a = e and ln(e) = 1 recovers the natural forms.

Combining rules in one problem

Real problems layer rules together. A function like x²·e^(3x) needs the product rule with a chain rule inside one factor. The strategy is to identify the outermost structure first (is it a product? a quotient? a composite?), apply that rule, and handle each piece — using whatever additional rule that piece requires — as you come to it.

Exponential and log derivatives

d/dx[eˣ] = eˣ · d/dx[ln x] = 1/x · d/dx[aˣ] = aˣ·ln a

With the chain rule: d/dx[e^u] = e^u·u′ and d/dx[ln u] = u′/u.

Worked example

Differentiate y = x²·e^(3x).

1.Recognize a product of f = x² and g = e^(3x), so use the product rule f′g + fg′.
2.Derivative of f: f′ = 2x.
3.Derivative of g by the chain rule: g′ = e^(3x)·3 = 3e^(3x).
4.Assemble: 2x·e^(3x) + x²·3e^(3x).
5.Factor out the common x·e^(3x): x·e^(3x)(2 + 3x).

Answer: y′ = 2x·e^(3x) + 3x²·e^(3x) = x·e^(3x)(2 + 3x).

Watch out

d/dx[e^(3x)] is 3e^(3x), not e^(3x) or 3x·e^(3x). The exponential part stays intact and you multiply by the derivative of the exponent (3). Dropping that chain-rule factor is easy to do when the exponential looks "already done".

Checkpoint

What is d/dx[ln(x² + 1)]?

Checkpoint

What is the derivative of f(x) = e^(x²)?

On the exam

When several rules stack, name the outermost operation before computing anything. "This is a product, whose second factor is a composite" gives you a plan and keeps the chain-rule factors from getting lost in a long expression.

Answer the 2 checkpoints as you read.