LuoguP4893 GodFly的求导工具 题解

Description

题目链接

给定一个 $n$ 次整系数函数 $f(x)$,问 $f(x)$ 的 $k$ 阶导在 $x_i$ 处的导数。

$1\leq n \leq 100,k\leq n,m\leq 10,a_i\leq 10^5,x_i\leq 10^5$

Solution

很好奇这题是怎么评紫的(?

咳咳,对于这道题,你先要有亿简单高等数学知识。

  • $f(x)=x^p$ 的 $k$ 阶导函数为 $f^{(k)}(x)=\prod_{i=1}^k(p-i+1)x^{p-k}$
  • $f(x)=g(x)+h(x)$ 的 $k$ 阶导函数为 $f^{(k)}(x)=g^{(k)}(x)+h^{(k)}(x)$

证明略(

假设你已经熟知了以上结论,那么就可以愉快的把这道题切掉了。

直接暴力拆出每一项的系数,暴力进行求导,找到其 $k$ 阶导函数,对于每个询问,直接暴力带入即可。

因为 $a_i\leq 10^5,x_i \leq 10^5$ 可能会导致答案爆 $int$,所以你还要耐心地写一个高精套一个高精模板。

Code

#include<bits/stdc++.h>
#define Tp template<typename Ty>
#define Ts template<typename Ty,typename... Ar>
#define W while
#define I inline
#define RI register int
#define LL long long
#define Cn const
#define gc getchar
#define DD isdigit(c=gc())
#define pc(c) putchar((c))
#define min(x,y) ((x)<(y)?(x):(y))
#define max(x,y) ((x)>(y)?(x):(y))
using namespace std;
namespace Debug{
    Tp I void _debug(Cn char* f,Ty t){cerr<<f<<'='<<t<<endl;}
    Ts I void _debug(Cn char* f,Ty x,Ar... y){W(*f!=',') cerr<<*f++;cerr<<'='<<x<<",";_debug(f+1,y...);}
    Tp ostream& operator<<(ostream& os,Cn vector<Ty>& V){os<<"[";for(Cn auto& vv:V) os<<vv<<",";os<<"]";return os;}
    #define gdb(...) _debug(#__VA_ARGS__,__VA_ARGS__)
}using namespace Debug;
namespace FastIO{
    Tp I void read(Ty& x){char c;int f=1;x=0;W(!DD) f=c^'-'?1:-1;W(x=(x<<3)+(x<<1)+(c&15),DD);x*=f;}
    Ts I void read(Ty& x,Ar&... y){read(x),read(y...);}
    Tp I void write(Ty x){x<0&&(pc('-'),x=-x,0),x<10?(pc(x+'0'),0):(write(x/10),pc(x%10+'0'),0);}
    Tp I void writeln(Cn Ty& x){write(x),pc('\n');}
}using namespace FastIO;
Cn int N=110;
class Int{//Manchery的高精模板
    #define BASE 1000000000
    public:
    typedef long long value;
    void New(size_t l){
        if (a!=NULL)delete[] a;a=new value[l];
        len=1;a[0]=0;sign=1;
    }
    Int():a(NULL),base(BASE){New(1);}
    Int(value x):a(NULL),base(BASE){New(1);*this=x;}
    Int(value x,value _base):a(NULL),base(_base){New(1);*this=x;}
    Int(const Int &B):a(NULL),base(BASE){New(1);*this=B;}
    ~Int(){delete[] a;}
    Int& operator =(value x){
        size_t l=1;for (value x1=max(x,-x);x1>=base;++l,x1/=base);New(l);
        if (x<0)x=-x,sign=0;else sign=1;
        len=0;while (x)a[len++]=x%base,x/=base;
        if (!len)a[len++]=0;
        return *this;
    }
    Int& operator =(const Int &A){
        New(A.len);len=A.len;memcpy(a,A.a,sizeof(value)*len);
        base=A.base;sign=A.sign;return *this;
    }
    friend Int operator -(Int A){A.sign=1-A.sign;return A;}
    bool operator !(){if (len==1&&a[0]==0)return 1;else return 0;}
    friend Int operator +(Int A,Int B){
        if (A.sign!=B.sign){B.sign=1-B.sign;return A-B;}
        if (A.base!=B.base)
            if (A.base>B.base)B.set_base(A.base);
            else A.set_base(B.base);
        Int res;res.set_base(A.base); int len=A.len>B.len?A.len:B.len;
        res.New(len+1);res.sign=A.sign;
        memset(res.a,0,(len+1)*sizeof(value));
        for (int i=0;i<len;++i){
            if (i<A.len)res.a[i]+=A.a[i];
            if (i<B.len)res.a[i]+=B.a[i];
        }
        for (int i=0;i<len;++i)
            if (res.a[i]>=res.base)++res.a[i+1],res.a[i]-=res.base;
        if (res.a[len])res.len=len+1;else res.len=len;
        if (!res)res.sign=1;return res;
    }
    friend Int operator -(Int A,Int B){
        if (A.sign!=B.sign){B.sign=1-B.sign;return A+B;}
        if (A.base!=B.base)
            if (A.base>B.base)B.set_base(A.base);
            else A.set_base(B.base);
        if (small(A,B))swap(A,B),A.sign=1-A.sign;
        Int res;res.set_base(A.base); int len=A.len>B.len?A.len:B.len;
        res.New(len);res.sign=A.sign;
        memset(res.a,0,len*sizeof(value));
        for (int i=0;i<len;++i){
            if (i>=B.len)res.a[i]+=A.a[i];
            else res.a[i]+=A.a[i]-B.a[i];
            if (res.a[i]<0)res.a[i]+=res.base,--res.a[i+1];
        }
        while (len>1&&!res.a[len-1])--len;res.len=len;
        if (!res)res.sign=1;return res;
    }
    friend Int operator *(Int A,Int B){
        if (A.base!=B.base)
            if (A.base>B.base)B.set_base(A.base);
            else A.set_base(B.base);
        Int res;res.set_base(A.base); int len=A.len+B.len;
        res.New(len);res.sign=(A.sign==B.sign);
        memset(res.a,0,len*sizeof(value));
        for (int i=0;i<A.len;++i)
            for (int j=0;j<B.len;++j){
                res.a[i+j]+=A.a[i]*B.a[j];
                res.a[i+j+1]+=res.a[i+j]/res.base;
                res.a[i+j]%=res.base;
            }
        /*
        for (int i=0;i<A.len;++i)
            for (int j=0;j<B.len;++j)res.a[i+j]+=A.a[i]*B.a[j];
        for (int i=0;i<len-1;++i)res.a[i+1]+=res.a[i]/res.base,res.a[i]%=res.base;
        */
        while (len>1&&!res.a[len-1])--len;res.len=len;
        return res;
    }
    friend pair<Int,Int> divide(Int A,Int B){
        if (!B){puts("error:div zero!");for (;;);}
        if (A.base!=B.base)
            if (A.base>B.base)B.set_base(A.base);
            else A.set_base(B.base);
        if (small(A,B))return make_pair(Int(0),A);
        Int C,D;C.set_base(A.base);D.set_base(A.base);C.New(A.len);C.len=A.len;
        bool Csign=(A.sign==B.sign),Dsign=A.sign;A.sign=B.sign=1;
        for (int i=A.len-1;i>=0;--i){
            C.a[i]=0;D=D*D.base;D.a[0]=A.a[i];
            int l=0,r=A.base-1,mid;
            while (l<r){
                mid=(l+r+1)>>1;
                if (small(B*mid,D+1))l=mid;
                    else r=mid-1;
            }
            C.a[i]=l;D=D-B*l;
        }
        C.sign=Csign;D.sign=Dsign;if (!D)D.sign=1;
        while (C.len>1&&!C.a[C.len-1])--C.len;
        return make_pair(C,D);
    }
    Int operator /(value x){
        if (!x){puts("error:div zero!");for (;;);}
        value d=0;Int res;res.set_base(base);res.New(len);res.len=len;
        if (x<0)x=-x,res.sign=(sign==0);
        else res.sign=(sign==1);
        for (int i=len-1;i>=0;--i)
            d=d*base+a[i],res.a[i]=d/x,d%=x;
        while (res.len>1&&!res.a[res.len-1])--res.len;
        return res;
    }
    Int operator %(value x){
        value d=0;if (x<0)x=-x;
        for (int i=len-1;i>=0;--i)d=(d*base+a[i])%x;
        return d;
    }
    friend Int abs(Int A){A.sign=1;return A;}
    friend bool small(Int A,Int B){
        if (A.base!=B.base)
            if (A.base>B.base)B.set_base(A.base);
            else A.set_base(B.base);
        if (A.len!=B.len)return A.len<B.len;
        for (int i=A.len-1;i>=0;--i)
            if (A.a[i]!=B.a[i])return A.a[i]<B.a[i];
        return 0;
    }
    friend bool operator <(Int A,Int B){
        if (A.sign!=B.sign)return A.sign<B.sign;
        return A.sign==1?small(A,B):small(B,A);
    }
    friend bool operator ==(Int A,Int B){
        if (A.base!=B.base)
            if (A.base>B.base)B.set_base(A.base);
            else A.set_base(B.base);
        if (A.sign!=B.sign||A.len!=B.len)return 0;
        for (int i=0;i<A.len;++i)if (A.a[i]!=B.a[i])return 0;
        return 1;
    }
    friend bool operator !=(Int A,Int B){return !(A==B);}
    friend bool operator >(Int A,Int B){return !(A<B||A==B);}
    friend bool operator <=(Int A,Int B){return A<B||A==B;}
    friend bool operator >=(Int A,Int B){return A>B||A==B;}
    Int operator /(Int B){return divide(*this,B).first;}
    Int operator %(Int B){return divide(*this,B).second;}
    Int& operator +=(Int B){*this=*this+B;return *this;}
    Int& operator -=(Int B){*this=*this-B;return *this;}
    Int& operator *=(Int B){*this=*this*B;return *this;}
    Int& operator /=(Int B){*this=*this/B;return *this;}
    Int& operator %=(Int B){*this=*this%B;return *this;}
    Int& operator ++(){*this=*this+1;return *this;}
    Int& operator --(){*this=*this-1;return *this;}
    Int operator ++(int){Int res(*this);*this=*this+1;return res;}
    Int operator --(int){Int res(*this);*this=*this-1;return res;}
    Int operator +(value x){return *this+Int(x,this->base);}
    Int operator -(value x){return *this-Int(x,this->base);}
    Int operator *(value x){return *this*Int(x,this->base);}
    //Int operator /(value x){Int T;T=x;return *this/T;}
    //Int operator %(value x){Int T;T=x;return *this%T;}
    Int& operator *=(value x){*this=*this*x;return *this;}
    Int& operator +=(value x){*this=*this+x;return *this;}
    Int& operator -=(value x){*this=*this-x;return *this;}
    Int& operator /=(value x){*this=*this/x;return *this;}
    Int& operator %=(value x){*this=*this%x;return *this;}
    bool operator ==(value x){return *this==Int(x,this->base);}
    bool operator !=(value x){return *this!=Int(x,this->base);}
    bool operator <=(value x){return *this<=Int(x,this->base);}
    bool operator >=(value x){return *this>=Int(x,this->base);}
    bool operator <(value x){return *this<Int(x,this->base);}
    bool operator >(value x){return *this>Int(x,this->base);}
    friend Int gcd(Int x,Int y){
        Int t;int cnt=0;
        while (1){
            if (x<y)t=x,x=y,y=t;
            if (y==0){
                while (cnt--)x*=2;
                return x;
            }
            if (x%2==0&&y%2==0)x/=2,y/=2,++cnt;
            else if (x%2==0)x/=2;
            else if (y%2==0)y/=2;
            else {t=x;x=y;y=t-y;}
        }
    }
    void to_arr(char *c){
        char *c1=c;
        for (int i=0;i<len-1;++i)
            for (value x=a[i],b=base/10;b>=1;b/=10)*c1++='0'+x%10,x/=10;
        for (value x=a[len-1];x>0;x/=10)*c1++='0'+x%10;
        if (len==1&&a[len]==0)*c1++='0';
        if (sign==0)*c1++='-';*c1=0;reverse(c,c1);
    }
    void from_arr(char *c){
        size_t base_l=get_basel(),b=1;int cl=strlen(c);value x=0;
        New((cl+base_l-1)/base_l);len=0;
        if (*c=='-')sign=0,++c,--cl;else sign=1;
        while (--cl>=0){
            x+=(c[cl]-'0')*b;b*=10;if (b==base)a[len++]=x,x=0,b=1;
        }
        if (!len||x)a[len++]=x;
        while (len>1&&!a[len-1])--len;
    }
    void set_base(int _base){
        if (base==_base)return;
        char *c=new char[len*get_basel()+1];
        to_arr(c);base=_base;from_arr(c);
        delete[] c;
    }
    void set_basel(int _l){
        size_t _base=1;while (_l--)_base*=10;set_base(_base);
    }
    void read(){
        vector<char> s;char ch;
        scanf(" %c",&ch);if (ch=='-')s.push_back('-'),ch=getchar();
        for (;ch>='0'&&ch<='9';ch=getchar())s.push_back(ch);
        char *c=new char[s.size()+1];
        for (int i=0;i<s.size();++i)c[i]=s[i];c[s.size()]=0;
        from_arr(c);delete[] c;
        if (!*this)this->sign=1;
    }
    void print(){
        if (!sign)putchar('-');
        printf("%d",int(a[len-1]));
        for (int i=len-2;i>=0;--i){
            for (int j=base/10;j>=10;j/=10)
                if (a[i]<j)putchar('0');
                    else break;
            printf("%d",(int)a[i]);
        }
    }
    void println(){print();putchar('\n');}
    private:
    value *a,base;int len;bool sign;  //0="-"
    size_t get_basel()const{
        size_t res=0;for (int b=base/10;b>=1;b/=10,++res);
        return res;
    }
    #undef BASE
};
Int num[N],Ans[N],top=1,ans;
int n,m,k,len;
string str;
I Int fpow(int a,int b){Int A=a,s=1;W(b!=0){if(b%2==1) s*=A;A*=A;b=b/2;}return s;}//高精快速幂
int main(){
    RI i,j,s;for(read(n,k),cin>>str,len=str.length(),i=5;i<len;)
        if(isdigit(str[i])){j=i+1;s=str[i]-'0';W(j<len&&isdigit(str[j])) s=s*10+str[j]-'0',j++;top=s,i=j;}//处理系数
        else if(str[i]=='x'){j=i+1;s=0;W(j<len&&str[j]!='^') ++j;j++;W(j<len&&isdigit(str[j])) s=s*10+str[j]-'0',++j;num[s]+=top;top=1;i=j;}//处理次数
        else i++;
    for(i=0;i<=n;i++)
        if(i-k>=0) for(Ans[i-k]=num[i],j=1;j<=k;j++) Ans[i-k]*=i-j+1;//直接求导套公式
    for(read(m);m--;){for(read(s),ans=j=0;j<=n;j++) ans=ans+Ans[j]*fpow(s,j);ans.println();}//对于每个询问直接暴力计算
    return 0;
}
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