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# diff

Differentiate symbolic expression or function

## Description

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diff(F) differentiates F with respect to the variable determined by symvar.

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diff(F,var) differentiates F with respect to the variable var.

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diff(F,n) computes the nth derivative of F with respect to the variable determined by symvar.

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diff(F,var,n) computes the nth derivative of F with respect to the variable var. This syntax is equivalent to diff(F,n,var).

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diff(F,var1,...,varN) differentiates F with respect to the variables var1,...,varN.

## Examples

### Differentiation of a Univariate Function

Find the first derivative of this univariate function:

```syms x
f(x) = sin(x^2);
df = diff(f)```
```df(x) =
2*x*cos(x^2)```

### Differentiation with Respect to a Particular Variable

Find the first derivative of this expression:

```syms x t
diff(sin(x*t^2))```
```ans =
t^2*cos(t^2*x)```

Because you did not specify the differentiation variable, diff uses the default variable defined by symvar. For this expression, the default variable is x:

`symvar(sin(x*t^2),1)`
```ans =
x```

Now, find the derivative of this expression with respect to the variable t:

`diff(sin(x*t^2),t)`
```ans =
2*t*x*cos(t^2*x)```

### Higher-Order Derivatives of a Univariate Expression

Find the 4th, 5th, and 6th derivatives of this expression:

```syms t
d4 = diff(t^6,4)
d5 = diff(t^6,5)
d6 = diff(t^6,6)```
```d4 =
360*t^2

d5 =
720*t

d6 =
720```

### Higher-Order Derivatives of a Multivariate Expression with Respect to a Particular Variable

Find the second derivative of this expression with respect to the variable y:

```syms x y
diff(x*cos(x*y), y, 2)```
```ans =
-x^3*cos(x*y)```

### Higher-Order Derivatives of a Multivariate Expression with Respect to the Default Variable

Compute the second derivative of the expression x*y. If you do not specify the differentiation variable, diff uses the variable determined by symvar. For this expression, symvar(x*y,1) returns x. Therefore, diff computes the second derivative of x*y with respect to x.

```syms x y
diff(x*y, 2)```
```ans =
0```

If you use nested diff calls and do not specify the differentiation variable, diff determines the differentiation variable for each call. For example, differentiate the expression x*y by calling the diff function twice:

`diff(diff(x*y))`
```ans =
1```

In the first call, diff differentiate x*y with respect to x, and returns y. In the second call, diff differentiates y with respect to y, and returns 1.

Thus, diff(x*y, 2) is equivalent to diff(x*y, x, x), and diff(diff(x*y)) is equivalent to diff(x*y, x, y).

### Mixed Derivatives

Differentiate this expression with respect to the variables x and y:

```syms x y
diff(x*sin(x*y), x, y)```
```ans =
2*x*cos(x*y) - x^2*y*sin(x*y)```

You also can compute mixed higher-order derivatives by providing all differentiation variables:

```syms x y
diff(x*sin(x*y), x, x, x, y)```
```ans =
x^2*y^3*sin(x*y) - 6*x*y^2*cos(x*y) - 6*y*sin(x*y)```

## Input Arguments

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### F — Expression or function to differentiatesymbolic expression | symbolic function | symbolic vector | symbolic matrix

Expression or function to differentiate, specified as a symbolic expression or function or as a vector or matrix of symbolic expressions or functions. If F is a vector or a matrix, diff differentiates each element of F and returns a vector or a matrix of the same size as F.

### var — Differentiation variablesymbolic variable | string

Differentiation variable, specified as a symbolic variable or a string.

### var1,...,varN — Differentiation variablessymbolic variables | strings

Differentiation variables, specified as symbolic variables or strings.

### n — Differentiation ordernonnegative integer

Differentiation order, specified as a nonnegative integer.

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### Tips

• When computing mixed higher-order derivatives, do not use n to specify the differentiation order. Instead, specify all differentiation variables explicitly.

• To improve performance, diff assumes that all mixed derivatives commute. For example,

$\frac{\partial }{\partial x}\frac{\partial }{\partial y}f\left(x,y\right)=\frac{\partial }{\partial y}\frac{\partial }{\partial x}f\left(x,y\right)$

This assumption suffices for most engineering and scientific problems.

• If you differentiate a multivariate expression or function F without specifying the differentiation variable, then a nested call to diff and diff(F,n) can return different results. This is because in a nested call, each differentiation step determines and uses its own differentiation variable. In calls like diff(F,n), the differentiation variable is determined once by symvar(F,1) and used for all differentiation steps.