Directional derivative is one of 3 multivariable formulas in the calculus section of this library, and it is used at university level.
Why directional derivative works
The gradient stores the rate along each axis and the unit vector says what share of your step goes along each axis, so the dot product combines them correctly. The result equals the gradient length times the cosine of the angle between the two, which is largest straight uphill and zero sideways.
What each symbol means
$\mathbf u$ is a unit direction vector.
Directional derivative: when it holds
$f$ must be differentiable and $\|\mathbf u\|=1$.
When it stops applying
The formula assumes the function is differentiable at the point. Along a crease, like the fold in the surface built from the absolute value of x, a one-sided rate in each direction still exists, but no single gradient vector reproduces all of them.
Directional derivative: a worked example
If $\nabla f=(2,3)$ and $\mathbf u=(1,0)$, then $D_{\mathbf u}f=2$.
The mistake to avoid
What people do: Dotting with the given direction without shrinking it to unit length first.
Why it goes wrong: A direction of length 5 multiplies the answer by 5, so what should be a rate per unit of distance turns into a number with no clear meaning.
Do this instead: Divide the direction vector by its own length before taking the dot product.
Directional derivative: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\mathbf u$ is a unit direction vector.
- Check the conditions before substituting. $f$ must be differentiable and $\|\mathbf u\|=1$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Multivariable
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where directional derivative comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Directional Derivative Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about directional derivative
Which direction gives the biggest value?
The gradient's own direction. There the cosine equals 1 and the rate is exactly the length of the gradient vector.
Can the answer be negative?
Yes. Any direction with a downhill component gives a negative rate, and the most negative possible is the direction opposite the gradient.
What if the direction is given as an angle?
Turn the angle into a unit vector using its cosine and sine, then take the dot product with the gradient as usual.
How does this relate to a partial derivative?
A partial derivative is the special case where the chosen unit vector points straight along one axis, so the dot product simply picks out that one component.