Math · Calculus I · Worked example

Differentiate a square root with the chain rule

Differentiate f(x) = √(x² + 9), then check the answer at x = 4.

f (x) = \sqrt{x^{2} + 9}

Write the root as a power

A square root is a power of one half: √u = u^(1/2). Written that way, the outside can be differentiated with the power rule. The inside is everything under the root.

f (x) = {(x^{2} + 9)}^{1 / 2}

Name the layers

The inside is u = x² + 9. The outside raises it to the power 1/2.

u = x^{2} + 9, f = u^{1 / 2}

Differentiate the outside

By the power rule, bring down the 1/2 and lower the exponent by one: 1/2 − 1 = −1/2. A negative exponent means a reciprocal, so u^(−1/2) = 1/√u.

\frac{d f}{d u} = \frac{1}{2} u^{- 1 / 2} = \frac{1}{2 \sqrt{u}}

Differentiate the inside

The derivative of x² + 9 is 2x. The constant 9 doesn’t change, so it contributes 0.

\frac{d u}{d x} = 2 x

Multiply, then put the inside back

Multiply the two rates and replace u with x² + 9.

f^{'} (x) = \frac{1}{2 \sqrt{x^{2} + 9}} \cdot 2 x

Simplify

The 2 in the numerator cancels the 2 in the denominator.

f^{'} (x) = \frac{x}{\sqrt{x^{2} + 9}}

Check with numbers

At x = 4, f(4) = √25 = 5, and the formula gives f′(4) = 4/√25 = 4/5 = 0.8. A small nudge agrees: f(4.001) = √25.008001 ≈ 5.00080, which is about 0.00080 more than f(4) for a step of 0.001, a rate of about 0.8.

\frac{4}{\sqrt{4^{2} + 9}} = \frac{4}{\sqrt{25}} = \frac{4}{5} = 0.8

Result

f′(x) = x/√(x² + 9), and f′(4) = 0.8.

f^{'} (x) = \frac{x}{\sqrt{x^{2} + 9}}

Your turn

Differentiate g(x) = √(4x − 3). For which x does the derivative exist?

Show the answer and explanation

g′(x) = 2/√(4x − 3), for x > 3/4.

Write g = u^(1/2) with u = 4x − 3. The outside gives 1/(2√u) and the inside gives 4, so g′(x) = 4/(2√(4x − 3)) = 2/√(4x − 3). The root needs 4x − 3 ≥ 0, and the derivative also needs a nonzero denominator, so g′(x) exists for x > 3/4.

\begin{array}{l} g (x) = \sqrt{4 x - 3} \\ g^{'} (x) = \frac{1}{2 \sqrt{4 x - 3}} \cdot 4 \\ g^{'} (x) = \frac{2}{\sqrt{4 x - 3}} \end{array}

Keep exploring

Open the steps in Math: the Checker marks both forms of f′(x) ✓, the unsimplified product and x/√(x² + 9).

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