(ctx, f, x, n=1, **options)
return +v
r"""
    Numerically computes the derivative of `f`, `f'(x)`, or generally for
    an integer `n \ge 0`, the `n`-th derivative `f^{(n)}(x)`.
    A few basic examples are::

        >>> from sympy.mpmath import *
        >>> mp.dps = 15; mp.pretty = True
        >>> diff(lambda x: x**2 + x, 1.0)
        3.0
        >>> diff(lambda x: x**2 + x, 1.0, 2)
        2.0
        >>> diff(lambda x: x**2 + x, 1.0, 3)
        0.0
        >>> nprint([diff(exp, 3, n) for n in range(5)])   # exp'(x) = exp(x)
        [20.0855, 20.0855, 20.0855, 20.0855, 20.0855]

    Even more generally, given a tuple of arguments `(x_1, \ldots, x_k)`
    and order `(n_1, \ldots, n_k)`, the partial derivative
    `f^{(n_1,\ldots,n_k)}(x_1,\ldots,x_k)` is evaluated. For example::

        >>> diff(lambda x,y: 3*x*y + 2*y - x, (0.25, 0.5), (0,1))
        2.75
        >>> diff(lambda x,y: 3*x*y + 2*y - x, (0.25, 0.5), (1,1))
        3.0

    **Options**

    The following optional keyword arguments are recognized:

    ``method``
        Supported methods are ``'step'`` or ``'quad'``: derivatives may be
        computed using either a finite difference with a small step
        size `h` (default), or numerical quadrature.
    ``direction``
        Direction of finite difference: can be -1 for a left
        difference, 0 for a central difference (default), or +1
        for a right difference; more generally can be any complex number.
    ``addprec``
        Extra precision for `h` used to account for the function's
        sensitivity to perturbations (default = 10).
    ``relative``
        Choose `h` relative to the magnitude of `x`, rather than an
        absolute value; useful for large or tiny `x` (default = False).
    ``h``
        As an alternative to ``addprec`` and ``relative``, manually
        select the step size `h`.
    ``singular``
        If True, evaluation exactly at the point `x` is avoided; this is
        useful for differentiating functions with removable singularities.
        Default = False.
    ``radius``
        Radius of integration contour (with ``method = 'quad'``).
        Default = 0.25. A larger radius typically is faster and more
        accurate, but it must be chosen so that `f` has no
        singularities within the radius from the evaluation point.

    A finite difference requires `n+1` function evaluations and must be
    performed at `(n+1)` times the target precision. Accordingly, `f` must
    support fast evaluation at high precision.

    With integration, a larger number of function evaluations is
    required, but not much extra precision is required. For high order
    derivatives, this method may thus be faster if f is very expensive to
    evaluate at high precision.

    **Further examples**

    The direction option is useful for computing left- or right-sided
    derivatives of nonsmooth functions::

        >>> diff(abs, 0, direction=0)
        0.0
        >>> diff(abs, 0, direction=1)
        1.0
        >>> diff(abs, 0, direction=-1)
        -1.0

    More generally, if the direction is nonzero, a right difference
    is computed where the step size is multiplied by sign(direction).
    For example, with direction=+j, the derivative from the positive
    imaginary direction will be computed::

        >>> diff(abs, 0, direction=j)
        (0.0 - 1.0j)

    With integration, the result may have a small imaginary part
    even even if the result is purely real::

        >>> diff(sqrt, 1, method='quad')    # doctest:+ELLIPSIS
        (0.5 - 4.59...e-26j)
        >>> chop(_)
        0.5

    Adding precision to obtain an accurate value::

        >>> diff(cos, 1e-30)
        0.0
        >>> diff(cos, 1e-30, h=0.0001)
        -9.99999998328279e-31
        >>> diff(cos, 1e-30, addprec=100)
        -1.0e-30
r"""
    Numerically computes the derivative of `f`, `f'(x)`, or generally for
    an integer `n \ge 0`, the `n`-th derivative `f^{(n)}(x)`.
    A few basic examples are::
r Numerically computes the derivative of f f ( x ) or generally for an integer n \ ge 0 the n - th derivative f^ { ( n ) } ( x ) . A few basic examples are ::
def diff(ctx, f, x, n=1, **options):
    r"""
    Numerically computes the derivative of `f`, `f'(x)`, or generally for
    an integer `n \ge 0`, the `n`-th derivative `f^{(n)}(x)`.
    A few basic examples are::

        >>> from sympy.mpmath import *
        >>> mp.dps = 15; mp.pretty = True
        >>> diff(lambda x: x**2 + x, 1.0)
        3.0
        >>> diff(lambda x: x**2 + x, 1.0, 2)
        2.0
        >>> diff(lambda x: x**2 + x, 1.0, 3)
        0.0
        >>> nprint([diff(exp, 3, n) for n in range(5)])   # exp'(x) = exp(x)
        [20.0855, 20.0855, 20.0855, 20.0855, 20.0855]

    Even more generally, given a tuple of arguments `(x_1, \ldots, x_k)`
    and order `(n_1, \ldots, n_k)`, the partial derivative
    `f^{(n_1,\ldots,n_k)}(x_1,\ldots,x_k)` is evaluated. For example::

        >>> diff(lambda x,y: 3*x*y + 2*y - x, (0.25, 0.5), (0,1))
        2.75
        >>> diff(lambda x,y: 3*x*y + 2*y - x, (0.25, 0.5), (1,1))
        3.0

    **Options**

    The following optional keyword arguments are recognized:

    ``method``
        Supported methods are ``'step'`` or ``'quad'``: derivatives may be
        computed using either a finite difference with a small step
        size `h` (default), or numerical quadrature.
    ``direction``
        Direction of finite difference: can be -1 for a left
        difference, 0 for a central difference (default), or +1
        for a right difference; more generally can be any complex number.
    ``addprec``
        Extra precision for `h` used to account for the function's
        sensitivity to perturbations (default = 10).
    ``relative``
        Choose `h` relative to the magnitude of `x`, rather than an
        absolute value; useful for large or tiny `x` (default = False).
    ``h``
        As an alternative to ``addprec`` and ``relative``, manually
        select the step size `h`.
    ``singular``
        If True, evaluation exactly at the point `x` is avoided; this is
        useful for differentiating functions with removable singularities.
        Default = False.
    ``radius``
        Radius of integration contour (with ``method = 'quad'``).
        Default = 0.25. A larger radius typically is faster and more
        accurate, but it must be chosen so that `f` has no
        singularities within the radius from the evaluation point.

    A finite difference requires `n+1` function evaluations and must be
    performed at `(n+1)` times the target precision. Accordingly, `f` must
    support fast evaluation at high precision.

    With integration, a larger number of function evaluations is
    required, but not much extra precision is required. For high order
    derivatives, this method may thus be faster if f is very expensive to
    evaluate at high precision.

    **Further examples**

    The direction option is useful for computing left- or right-sided
    derivatives of nonsmooth functions::

        >>> diff(abs, 0, direction=0)
        0.0
        >>> diff(abs, 0, direction=1)
        1.0
        >>> diff(abs, 0, direction=-1)
        -1.0

    More generally, if the direction is nonzero, a right difference
    is computed where the step size is multiplied by sign(direction).
    For example, with direction=+j, the derivative from the positive
    imaginary direction will be computed::

        >>> diff(abs, 0, direction=j)
        (0.0 - 1.0j)

    With integration, the result may have a small imaginary part
    even even if the result is purely real::

        >>> diff(sqrt, 1, method='quad')    # doctest:+ELLIPSIS
        (0.5 - 4.59...e-26j)
        >>> chop(_)
        0.5

    Adding precision to obtain an accurate value::

        >>> diff(cos, 1e-30)
        0.0
        >>> diff(cos, 1e-30, h=0.0001)
        -9.99999998328279e-31
        >>> diff(cos, 1e-30, addprec=100)
        -1.0e-30

    """
    partial = False
    try:
        orders = list(n)
        x = list(x)
        partial = True
    except TypeError:
        pass
    if partial:
        x = [ctx.convert(_) for _ in x]
        return _partial_diff(ctx, f, x, orders, options)
    method = options.get('method', 'step')
    if n == 0 and method != 'quad' and not options.get('singular'):
        return f(ctx.convert(x))
    prec = ctx.prec
    try:
        if method == 'step':
            values, norm, workprec = hsteps(ctx, f, x, n, prec, **options)
            ctx.prec = workprec
            v = ctx.difference(values, n) / norm**n
        elif method == 'quad':
            ctx.prec += 10
            radius = ctx.convert(options.get('radius', 0.25))
            def g(t):
                rei = radius*ctx.expj(t)
                z = x + rei
                return f(z) / rei**n
            d = ctx.quadts(g, [0, 2*ctx.pi])
            v = d * ctx.factorial(n) / (2*ctx.pi)
        else:
            raise ValueError("unknown method: %r" % method)
    finally:
        ctx.prec = prec
    return +v
def diff ( ctx , f , x , n = 1 , * * options ) : partial = False try : orders = list ( n ) x = list ( x ) partial = True except TypeError : pass if partial : x = [ ctx . convert ( _ ) for _ in x ] return _partial_diff ( ctx , f , x , orders , options ) method = options . get ( 'method' , 'step' ) if n == 0 and method != 'quad' and not options . get ( 'singular' ) : return f ( ctx . convert ( x ) ) prec = ctx . prec try : if method == 'step' : values , norm , workprec = hsteps ( ctx , f , x , n , prec , * * options ) ctx . prec = workprec v = ctx . difference ( values , n ) / norm ** n elif method == 'quad' : ctx . prec += 10 radius = ctx . convert ( options . get ( 'radius' , 0.25 ) ) def g ( t ) : rei = radius * ctx . expj ( t ) z = x + rei return f ( z ) / rei ** n d = ctx . quadts ( g , [ 0 , 2 * ctx . pi ] ) v = d * ctx . factorial ( n ) / ( 2 * ctx . pi ) else : raise ValueError ( "unknown method: %r" % method ) finally : ctx . prec = prec return + v