# What is the difference between the gradient and the directional derivative?

Consider a function $$g: \mathbb{R^3} \to \mathbb{R}$$ defined by $$g(x,y,z) = z^2 -x^3 + 2z + 3y^3$$

Find the gradient of $$g$$ at the point $$(2,1,-1)$$.

Is the gradient $$\nabla g(2,1,-1)$$ given by a vector, that is, $$\nabla g(2,1,-1) = -12i + 9j$$? If so, then what does the directional derivative mean?

The gradient is a vector; it points in the direction of steepest ascent.

The directional derivative is a number; it is the rate of change when your point in $\Bbb R^3$ moves in that direction. (You can imagine "reducing" your function to a function of a single variable, say $t$, by "slicing" the curve in that direction; the directional derivative is just then the 1-D derivative of that "sliced" function.)

• steepest ascent, does that mean that for example $\nabla f = (a,b)$ gives us the fastest growing direction $(a,b)$. And from this that we should increase $x$ and $y$ with a scalar times $a$ and $b$ respectively in order to increase the function value $f$ as fast as possible? Commented May 22, 2018 at 15:37

Be careful that directional derivative of a function is a scalar while gradient is a vector.

The only difference between derivative and directional derivative is the definition of those terms. Remember:

• Directional derivative is the instantaneous rate of change (which is a scalar) of $f(x,y)$ in the direction of the unit vector $u$.
• Derivative is the rate of change of $f(x,y)$, which can be thought of the slope of the function at a point $(x_0,y_0)$.

Yes, the gradient is given by the row vector whose elements are the partial derivatives of $g$ with respect to $x$, $y$, and $z$, respectively. In your case the gradient at $(x,y,z)$ is hence $[-3x^2,9y^2,2z+2]$. The gradient at $(2,1,-1)$ is therefore $[-12,9,0]$.

The directional derivative at a point $(x,y,z)$ in direction $(u,v,w)$ is the gradient multiplied by the direction divided by its length. So if $u^2+v^2+w^2=1$ then the directional derivative at $(x,y,z)$ in direction $(u,v,w)$ is just $-3x^2 u + 9y^2v + (2z+2)w$.

If $u^2+v^2+w^2\neq 1$ then you should divide the number above by $\sqrt{u^2+v^2+w^2}$.

In sum, the gradient is a vector with the slope of the function along each of the coordinate axes whereas the directional derivative is the slope in an arbitrary specified direction.

• A Directional Derivative is a value which represents a rate of change
• A Gradient is an angle/vector which points to the direction of the steepest ascent of a curve.

Let us take a look at the plot of the following function:

$$\bbox[lightgray] {f(x) = -x^2+4}\qquad (1)$$

The 1st derivative of the function is:

$$\bbox[lightgray] {\frac{dy}{dx} = -2x}\qquad (2)$$

Putting $$x=-1$$ in $$(2)$$ we obtain,

$$\implies \frac{dy}{dx} = \frac{rise}{run}= 2 ... ... ...\qquad (3)$$

Also,

$$tan \theta = 2$$
$$\implies \theta = tan^{-1}(2)$$
$$\implies \theta = 0.964$$ radian

So, $$\theta = 55.23 ^\circ ... ... ...\qquad (4)$$

.

Similarly, putting $$x=-2$$ in $$(2)$$ we obtain,

$$\implies \frac{dy}{dx} = \frac{rise}{run}= 4 ... ... ...\qquad (5)$$

So, $$\theta = 57.25 ^\circ ... ... ...\qquad (6)$$

• (3) and (5) are Directional Derivatives.
• Directional Derivatives are scalar values.

And,

• (4) and (6) are Gradients.
• Very good illustration. Makes the point clearly. Commented Mar 11, 2020 at 8:06
• This is an old question, yet this incorrect answer still has +6 upvote. The context of the question is multivariable calculus, and this answer does not make sense in that context.
– user26672
Commented Mar 4, 2021 at 4:53
• So in deep learning, when we calculate the gradient, are we actually calculating gradient or derivative? Because we are calculating by how much we should move in a certain direction for optimization, right? Commented Jul 13, 2021 at 18:14
• How does (6) 𝜃=57.25... represent a vector value? Looks like a scalar to me... Commented Mar 29, 2023 at 4:12

Taking a function $$g: \mathbb{R}^3\rightarrow \mathbb{R}$$. Its gradient $$\nabla{g}$$ is a vector, while the directional derivative is the inner product of another vector $$\mathbf{v}$$ in the same space with $$\nabla g$$, denoted as $$\mathbf{v}\cdot\nabla g$$. It means the projection of the vector $$\nabla g$$ onto the direction of the vector $$\mathbf{v}$$, which represents how the function $$g$$ varies in that direction.

In simple words, directional derivative can be visualized as slope of the function at the given point along a particular direction. For example partial derivative w.r.t x of a function can also be written as directional derivative of that function along x direction.

Gradient is a vector and for a given direction, directional derivative can be written as projection of gradient along that direction.